diff options
author | Norbert Preining <norbert@preining.info> | 2019-09-02 13:46:59 +0900 |
---|---|---|
committer | Norbert Preining <norbert@preining.info> | 2019-09-02 13:46:59 +0900 |
commit | e0c6872cf40896c7be36b11dcc744620f10adf1d (patch) | |
tree | 60335e10d2f4354b0674ec22d7b53f0f8abee672 /graphics/pstricks/contrib/pst-func/doc |
Initial commit
Diffstat (limited to 'graphics/pstricks/contrib/pst-func/doc')
-rw-r--r-- | graphics/pstricks/contrib/pst-func/doc/pst-func-doc.bib | 168 | ||||
-rw-r--r-- | graphics/pstricks/contrib/pst-func/doc/pst-func-doc.data | 4560 | ||||
-rw-r--r-- | graphics/pstricks/contrib/pst-func/doc/pst-func-doc.pdf | bin | 0 -> 3956145 bytes | |||
-rw-r--r-- | graphics/pstricks/contrib/pst-func/doc/pst-func-doc.tex | 2580 |
4 files changed, 7308 insertions, 0 deletions
diff --git a/graphics/pstricks/contrib/pst-func/doc/pst-func-doc.bib b/graphics/pstricks/contrib/pst-func/doc/pst-func-doc.bib new file mode 100644 index 0000000000..e749f71bfb --- /dev/null +++ b/graphics/pstricks/contrib/pst-func/doc/pst-func-doc.bib @@ -0,0 +1,168 @@ +@STRING{tugboat = {TUGboat} } +@STRING{beiprogramm = {{\TeX}-Beiprogramm} } +@STRING{bretter = {Bretter, die die Welt bedeuten} } +@STRING{dtk = {{D}ie {\TeX}nische {K}om{\"o}die} } +@STRING{editorial = {Editorial} } +@STRING{fremdebuehne = {Von fremden B{\"u}hnen} } +@STRING{fundus = {Aus dem Fundus} } +@STRING{hinterbuehne = {Hinter der B{\"u}hne} } +@STRING{leserbrief = {Leserbrief(e)} } +@STRING{magazin = {Magazin} } +@STRING{rezension = {Rezensionen} } +@STRING{schonimmer = {Was Sie schon immer {\"u}ber {\TeX} wissen wollten \dots} } +@STRING{theaterkasse = {Von der Theaterkasse} } +@STRING{theatertage = {{\TeX}-Theatertage} } + +@Article{ dtk02.2:jackson.voss:plot-funktionen, + author = {Laura E. Jackson and Herbert Voß}, + title = {Die {P}lot-{F}unktionen von {\texttt{pst-plot}}}, + journal = dtk, + year = 2002, + volume = {2/02}, + altvolume = 2, + altnumber = 14, + month = jun, + pages = {27--34}, + annote = bretter, + keywords = {}, + abstract = { Im letzten Heft wurden die mathematischen Funktionen von + \PS~im Zusammenhang mit dem {\LaTeX}-Paket + \texttt{pst-plot} zum Zeichnen von Funktionen beschrieben + und durch Beispiele erl{\"a}utert. In diesem Teil werden + die bislang nur erw{\"a}hnten Plot-Funktionen f{\"u}r + externe Daten behandelt. } +} + +@Article{ dtk02.1:voss:mathematischen, + author = {Herbert Voß}, + title = {Die mathematischen {F}unktionen von {P}ost{S}cript}, + journal = dtk, + year = 2002, + volume = {1/02}, + altvolume = 1, + altnumber = 14, + month = mar, + pages = {}, + annote = bretter, + keywords = {}, + abstract = { \PS, faktisch genauso alt wie {\TeX}, ist im + Verh{\"a}ltnis dazu allgemein noch weniger bekannt, wenn es + darum geht zu beurteilen, was es denn nun im eigentlichen + Sinne ist. Außerdem wird h{\"a}ufig vergessen, dass + sich mit den \PS-Funktionen viele Dinge erledigen lassen, + bei denen sonst auf externe Programme zur{\"u}ckgegriffen + wird. Dies wird im Folgenden f{\"u}r die mathematischen + Funktionen im Zusammenhang mit dem Paket \texttt{pst-plot} + gezeigt. } +} + +@Book{tlgc2, + author = {Michel Goosens and Frank Mittelbach and Sebastian Rahtz and Denis Roegel and Herbert Voß}, + title = {The {\LaTeX} {G}raphics {C}ompanion}, + publisher = {{Addison-Wesley Publishing Company}}, + edition = 2, + year = {2007}, + address = {Reading, Mass.} +} + + +@Article{girou:01:, + author = {Denis Girou}, + title = {Pr\'esentation de {PST}ricks}, + journal = {Cahier {GUT}enberg}, + year = 1994, + volume = {16}, + month = apr, + pages = {21-70} +} + +@Article{girou:02:, + author = {{Timothy van} Zandt and Denis Girou}, + title = {Inside {PST}ricks}, + journal = TUGboat, + year = 1994, + volume = {15}, + month = sep, + pages = {239-246} +} + +@Book{PostScript, + Author = {Kollock, Nikolai G.}, + Title = {Post{S}cript richtig eingesetzt: vom {K}onzept zum + praktischen {E}insatz}, + Publisher = {IWT}, + Address = {Vaterstetten}, + year = 1989, +} + +@online{pstricks, + Title = {PSTricks - {\PS} macros for generic {\TeX}}, + Author = {{Timothy van} Zandt}, + Organization = {}, + url = {http://www.tug.org/application/PSTricks}, + year = 1993 +} + +@ctan{pst-plot, + Title = {\texttt{pst-plot}: Plotting two dimensional functions and data}, + Author = {{Timothy van} Zandt and Herbert Voß}, + Organization = {}, + url = {/graphics/pstricks/generic/pst-plot.tex}, + year = 1999 +} + +@ctan{multido, + Title = {\texttt{multido.tex} - a loop macro, that supports fixed-point addition}, + Author = {{Timothy van} Zandt}, + url = {/graphics/pstricks/generic/multido.tex}, + Note = {}, + year = 1997 +} + +@Book{PSTricks2, + author = {Herbert Voß}, + title = {\texttt{PSTricks} -- {G}rafik f\"ur \TeX{} und \LaTeX}, + edition = {7}, + publisher = {DANTE -- Lehmanns}, + year = {2016}, + publisher = {Heidelberg and Berlin} +} + +@Book{voss:math, + author = {Herbert Voß}, + title = {Typesetting mathematics with \LaTeX}, + publisher = {UIT}, + year = {2010}, + address = {Cambridge} +} + +@Book{PSTricks2-UIT, + author = {Herbert Voß}, + title = {PSTricks -- Graphics for \TeX\ and \LaTeX}, + publisher = {UIT}, + year = {2011}, + address = {Cambridge} +} + +@Book{LaTeXRef-UIT, + author = {Herbert Voß}, + title = {{\LaTeX} quick reference}, + publisher = {UIT}, + year = {2012}, + address = {Cambridge} +} + +@online{wolfram, + author = {Eric Weisstein}, + title = {Wolfram MathWorld}, + publisher = {{Wolfram}}, + year = {2007}, + url = {http://mathworld.wolfram.com} +} + +@ctan{pst-tools, + author = {Herbert Voß}, + title = {\texttt{pst-tools} -- Helper functions}, + year = {2012}, + url = {/graphics/pstricks/contrib/pst-tools} +} diff --git a/graphics/pstricks/contrib/pst-func/doc/pst-func-doc.data b/graphics/pstricks/contrib/pst-func/doc/pst-func-doc.data new file mode 100644 index 0000000000..c39f6cddfc --- /dev/null +++ b/graphics/pstricks/contrib/pst-func/doc/pst-func-doc.data @@ -0,0 +1,4560 @@ +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +-1.57454 0.919992 +-1.57454 1.09572 +] +[ +-1.56752 0.856729 +-1.56752 1.15195 +] +[ +-1.56049 0.814554 +-1.56049 1.19413 +] +[ +-1.55346 0.779408 +-1.55346 1.22225 +] +[ +-1.54643 0.744262 +-1.54643 1.25036 +] +[ +-1.5394 0.716146 +-1.5394 1.27848 +] +[ +-1.53237 0.695058 +-1.53237 1.29957 +] +[ +-1.52534 0.666941 +-1.52534 1.32065 +] +[ +-1.51831 0.645854 +-1.51831 1.33471 +] +[ +-1.51128 0.624766 +-1.51128 1.3558 +] +[ +-1.50425 0.603679 +-1.50425 1.36986 +] +[ +-1.49722 0.582591 +-1.49722 1.38392 +] +[ +-1.49019 0.561504 +-1.49019 1.39798 +] +[ +-1.48316 0.547446 +-1.48316 1.41203 +] +[ +-1.47614 0.526358 +-1.47614 1.42609 +] +[ +-1.46911 0.5123 +-1.46911 1.44015 +] +[ +-1.46208 0.498241 +-1.46208 1.44718 +] +[ +-1.45505 0.477154 +-1.45505 1.46124 +] +[ +-1.44802 0.463095 +-1.44802 1.46827 +] +[ +-1.44099 0.449037 +-1.44099 1.48233 +] +[ +-1.43396 0.434979 +-1.43396 1.48935 +] +[ +-1.42693 0.42092 +-1.42693 1.50341 +] +[ +-1.4199 0.406862 +-1.4199 1.51044 +] +[ +-1.41287 0.392803 +-1.41287 1.51747 +] +[ +-1.40584 0.378745 +-1.40584 1.53153 +] +[ +-1.39881 0.364687 +-1.39881 1.53856 +] +[ +-1.39178 0.350628 +-1.39178 1.54559 +] +[ +-1.38476 0.33657 +-1.38476 1.55262 +] +[ +-1.37773 0.329541 +-1.37773 1.55965 +] +[ +-1.3707 0.315482 +-1.3707 1.56668 +] +[ +-1.36367 0.301424 +-1.36367 1.5737 +] +[ +-1.35664 0.287365 +-1.35664 1.58073 +] +[ +-1.34961 0.280336 +-1.34961 1.58776 +] +[ +-1.34258 0.266278 +-1.34258 1.59479 +] +[ +-1.33555 0.252219 +-1.33555 1.60182 +] +[ +-1.32852 0.24519 +-1.32852 1.60885 +] +[ +-1.32149 0.231132 +-1.32149 1.61588 +] +[ +-1.31446 0.224103 +-1.31446 1.62291 +] +[ +-1.30743 0.210044 +-1.30743 1.62994 +] +[ +-1.3004 0.203015 +-1.3004 1.63697 +] +[ +-1.29338 0.188957 +-1.29338 1.63697 +] +[ +-1.28635 0.181927 +-1.28635 1.644 +] +[ +-1.27932 0.167869 +-1.27932 1.65103 +] +[ +-1.27229 0.16084 +-1.27229 1.65805 +] +[ +-1.26526 0.146781 +-1.26526 1.65805 +] +[ +-1.25823 0.139752 +-1.25823 1.66508 +] +[ +-1.2512 0.125694 +-1.2512 1.67211 +] +[ +-1.24417 0.118665 +-1.24417 1.67914 +] +[ +-1.23714 0.111635 +-1.23714 1.67914 +] +[ +-1.23011 0.0975769 +-1.23011 1.68617 +] +[ +-1.22308 0.0905477 +-1.22308 1.68617 +] +[ +-1.21605 0.0835185 +-1.21605 1.6932 +] +[ +-1.20902 0.0694601 +-1.20902 1.70023 +] +[ +-1.202 0.0624309 +-1.202 1.70023 +] +[ +-1.19497 0.0554017 +-1.19497 1.70726 +] +[ +-1.18794 0.0413433 +-1.18794 1.70726 +] +[ +-1.18091 0.0343141 +-1.18091 1.71429 +] +[ +-1.17388 0.0272849 +-1.17388 1.71429 +] +[ +-1.16685 0.0202557 +-1.16685 1.72132 +] +[ +-1.15982 0.0061973 +-1.15982 1.72132 +] +[ +-1.15279 -0.000831896 +-1.15279 1.72835 +] +[ +-1.14576 -0.0078611 +-1.14576 1.72835 +] +[ +-1.13873 -0.0148903 +-1.13873 1.73538 +] +[ +-1.1317 -0.0289487 +-1.1317 1.73538 +] +[ +-1.12467 -0.0359779 +-1.12467 1.7424 +] +[ +-1.11764 -0.0430071 +-1.11764 1.7424 +] +[ +-1.11062 -0.0500363 +-1.11062 1.74943 +] +[ +-1.10359 -0.0570655 +-1.10359 1.74943 +] +[ +-1.09656 -0.0711239 +-1.09656 1.74943 +] +[ +-1.08953 -0.0781531 +-1.08953 1.75646 +] +[ +-1.0825 -0.0851823 +-1.0825 1.75646 +] +[ +-1.07547 -0.0922115 +-1.07547 1.76349 +] +[ +-1.06844 -0.0992407 +-1.06844 1.76349 +] +[ +-1.06141 -0.10627 +-1.06141 1.76349 +] +[ +-1.05438 -0.113299 +-1.05438 1.77052 +] +[ +-1.04735 -0.120328 +-1.04735 1.77052 +] +[ +-1.04032 -0.134387 +-1.04032 1.77052 +] +[ +-1.03329 -0.141416 +-1.03329 1.77755 +] +[ +-1.02626 -0.148445 +-1.02626 1.77755 +] +[ +-1.01924 -0.155474 +-1.01924 1.77755 +] +[ +-1.01221 -0.162504 +-1.01221 1.77755 +] +[ +-1.00518 -0.169533 +-1.00518 1.78458 +] +[ +-0.998147 -0.176562 +-0.998147 1.78458 +] +[ +-0.991118 -0.183591 +-0.991118 1.78458 +] +[ +-0.984089 -0.19062 +-0.984089 1.78458 +] +[ +-0.97706 -0.19765 +-0.97706 1.79161 +] +[ +-0.970031 -0.204679 +-0.970031 1.79161 +] +[ +-0.963001 -0.211708 +-0.963001 1.79161 +] +[ +-0.955972 -0.218737 +-0.955972 1.79161 +] +[ +-0.948943 -0.225766 +-0.948943 1.79864 +] +[ +-0.941914 -0.232796 +-0.941914 1.79864 +] +[ +-0.934885 -0.239825 +-0.934885 1.79864 +] +[ +-0.927856 -0.246854 +-0.927856 1.79864 +] +[ +-0.920826 -0.253883 +-0.920826 1.79864 +] +[ +-0.913797 -0.260912 +-0.913797 1.79864 +] +[ +-0.906768 -0.267942 +-0.906768 1.80567 +] +[ +-0.899739 -0.274971 +-0.899739 1.80567 +] +[ +-0.89271 -0.282 +-0.89271 1.80567 +] +[ +-0.88568 -0.289029 +-0.88568 1.80567 +] +[ +-0.878651 -0.296058 +-0.878651 1.80567 +] +[ +-0.871622 -0.303088 +-0.871622 1.80567 +] +[ +-0.864593 -0.310117 +-0.864593 1.80567 +] +[ +-0.857564 -0.317146 +-0.857564 1.80567 +] +[ +-0.850535 -0.324175 +-0.850535 1.80567 +] +[ +-0.843506 -0.331204 +-0.843506 1.80567 +] +[ +-0.836476 -0.338234 +-0.836476 1.8127 +] +[ +-0.829447 -0.345263 +-0.829447 1.8127 +] +[ +-0.822418 -0.352292 +-0.822418 1.8127 +] +[ +-0.815389 -0.359321 +-0.815389 1.8127 +] +[ +-0.80836 -0.36635 +-0.80836 1.8127 +] +[ +-0.80133 -0.36635 +-0.80133 1.8127 +] +[ +-0.794301 -0.37338 +-0.794301 1.8127 +] +[ +-0.787272 -0.380409 +-0.787272 1.8127 +] +[ +-0.780243 -0.387438 +-0.780243 1.8127 +] +[ +-0.773214 -0.394467 +-0.773214 1.8127 +] +[ +-0.766185 -0.401496 +-0.766185 1.8127 +] +[ +-0.759155 -0.408526 +-0.759155 1.8127 +] +[ +-0.752126 -0.415555 +-0.752126 1.8127 +] +[ +-0.745097 -0.422584 +-0.745097 1.8127 +] +[ +-0.738068 -0.429613 +-0.738068 1.8127 +] +[ +-0.731039 -0.436642 +-0.731039 1.8127 +] +[ +-0.72401 -0.436642 +-0.72401 1.8127 +] +[ +-0.71698 -0.443672 +-0.71698 1.8127 +] +[ +-0.709951 -0.450701 +-0.709951 1.80567 +] +[ +-0.702922 -0.45773 +-0.702922 1.80567 +] +[ +-0.695893 -0.464759 +-0.695893 1.80567 +] +[ +-0.688864 -0.471788 +-0.688864 1.80567 +] +[ +-0.681834 -0.478818 +-0.681834 1.80567 +] +[ +-0.674805 -0.485847 +-0.674805 1.80567 +] +[ +-0.667776 -0.485847 +-0.667776 1.80567 +] +[ +-0.660747 -0.492876 +-0.660747 1.80567 +] +[ +-0.653718 -0.499905 +-0.653718 1.80567 +] +[ +-0.646689 -0.506934 +-0.646689 1.80567 +] +[ +-0.639659 -0.513964 +-0.639659 1.79864 +] +[ +-0.63263 -0.520993 +-0.63263 1.79864 +] +[ +-0.625601 -0.528022 +-0.625601 1.79864 +] +[ +-0.618572 -0.528022 +-0.618572 1.79864 +] +[ +-0.611543 -0.535051 +-0.611543 1.79864 +] +[ +-0.604514 -0.54208 +-0.604514 1.79864 +] +[ +-0.597484 -0.549109 +-0.597484 1.79161 +] +[ +-0.590455 -0.556139 +-0.590455 1.79161 +] +[ +-0.583426 -0.563168 +-0.583426 1.79161 +] +[ +-0.576397 -0.563168 +-0.576397 1.79161 +] +[ +-0.569368 -0.570197 +-0.569368 1.79161 +] +[ +-0.562338 -0.577226 +-0.562338 1.78458 +] +[ +-0.555309 -0.584255 +-0.555309 1.78458 +] +[ +-0.54828 -0.591285 +-0.54828 1.78458 +] +[ +-0.541251 -0.598314 +-0.541251 1.78458 +] +[ +-0.534222 -0.605343 +-0.534222 1.77755 +] +[ +-0.527193 -0.605343 +-0.527193 1.77755 +] +[ +-0.520163 -0.612372 +-0.520163 1.77755 +] +[ +-0.513134 -0.619401 +-0.513134 1.77755 +] +[ +-0.506105 -0.62643 +-0.506105 1.77052 +] +[ +-0.499076 -0.63346 +-0.499076 1.77052 +] +[ +-0.492047 -0.640489 +-0.492047 1.77052 +] +[ +-0.485017 -0.640489 +-0.485017 1.76349 +] +[ +-0.477988 -0.647518 +-0.477988 1.76349 +] +[ +-0.470959 -0.654547 +-0.470959 1.76349 +] +[ +-0.46393 -0.661576 +-0.46393 1.75646 +] +[ +-0.456901 -0.668605 +-0.456901 1.75646 +] +[ +-0.449871 -0.675635 +-0.449871 1.75646 +] +[ +-0.442842 -0.675635 +-0.442842 1.74943 +] +[ +-0.435813 -0.682664 +-0.435813 1.74943 +] +[ +-0.428784 -0.689693 +-0.428784 1.74943 +] +[ +-0.421755 -0.696722 +-0.421755 1.7424 +] +[ +-0.414725 -0.703751 +-0.414725 1.7424 +] +[ +-0.407696 -0.71078 +-0.407696 1.73538 +] +[ +-0.400667 -0.71078 +-0.400667 1.73538 +] +[ +-0.393638 -0.71781 +-0.393638 1.73538 +] +[ +-0.386609 -0.724839 +-0.386609 1.72835 +] +[ +-0.379579 -0.731868 +-0.379579 1.72835 +] +[ +-0.37255 -0.738897 +-0.37255 1.72132 +] +[ +-0.365521 -0.745926 +-0.365521 1.72132 +] +[ +-0.358492 -0.745926 +-0.358492 1.71429 +] +[ +-0.351463 -0.752955 +-0.351463 1.71429 +] +[ +-0.344433 -0.759985 +-0.344433 1.70726 +] +[ +-0.337404 -0.767014 +-0.337404 1.70726 +] +[ +-0.330375 -0.774043 +-0.330375 1.70023 +] +[ +-0.323346 -0.781072 +-0.323346 1.70023 +] +[ +-0.316317 -0.788101 +-0.316317 1.6932 +] +[ +-0.309287 -0.79513 +-0.309287 1.6932 +] +[ +-0.302258 -0.79513 +-0.302258 1.68617 +] +[ +-0.295229 -0.80216 +-0.295229 1.67914 +] +[ +-0.2882 -0.809189 +-0.2882 1.67914 +] +[ +-0.281171 -0.816218 +-0.281171 1.67211 +] +[ +-0.274141 -0.823247 +-0.274141 1.67211 +] +[ +-0.267112 -0.830276 +-0.267112 1.66508 +] +[ +-0.260083 -0.837305 +-0.260083 1.65805 +] +[ +-0.253054 -0.844335 +-0.253054 1.65805 +] +[ +-0.246024 -0.851364 +-0.246024 1.65103 +] +[ +-0.238995 -0.858393 +-0.238995 1.644 +] +[ +-0.231966 -0.865422 +-0.231966 1.644 +] +[ +-0.224937 -0.872451 +-0.224937 1.63697 +] +[ +-0.217908 -0.879481 +-0.217908 1.62994 +] +[ +-0.210878 -0.879481 +-0.210878 1.62291 +] +[ +-0.203849 -0.88651 +-0.203849 1.62291 +] +[ +-0.19682 -0.893539 +-0.19682 1.61588 +] +[ +-0.189791 -0.900568 +-0.189791 1.60885 +] +[ +-0.182762 -0.914626 +-0.182762 1.60182 +] +[ +-0.175732 -0.921656 +-0.175732 1.59479 +] +[ +-0.168703 -0.928685 +-0.168703 1.58776 +] +[ +-0.161674 -0.935714 +-0.161674 1.58776 +] +[ +-0.154645 -0.942743 +-0.154645 1.58073 +] +[ +-0.147616 -0.949772 +-0.147616 1.5737 +] +[ +-0.140586 -0.956801 +-0.140586 1.56668 +] +[ +-0.133557 -0.963831 +-0.133557 1.55965 +] +[ +-0.126528 -0.97086 +-0.126528 1.55262 +] +[ +-0.119499 -0.984918 +-0.119499 1.54559 +] +[ +-0.11247 -0.991947 +-0.11247 1.53153 +] +[ +-0.10544 -0.998976 +-0.10544 1.5245 +] +[ +-0.0984112 -1.00601 +-0.0984112 1.51747 +] +[ +-0.091382 -1.02006 +-0.091382 1.51044 +] +[ +-0.0843528 -1.02709 +-0.0843528 1.50341 +] +[ +-0.0773236 -1.04115 +-0.0773236 1.48935 +] +[ +-0.0702944 -1.04818 +-0.0702944 1.48233 +] +[ +-0.0632652 -1.06224 +-0.0632652 1.46827 +] +[ +-0.056236 -1.06927 +-0.056236 1.46124 +] +[ +-0.0492068 -1.08333 +-0.0492068 1.44718 +] +[ +-0.0421776 -1.09738 +-0.0421776 1.43312 +] +[ +-0.0351484 -1.11144 +-0.0351484 1.41906 +] +[ +-0.0281192 -1.1255 +-0.0281192 1.405 +] +[ +-0.02109 -1.14659 +-0.02109 1.38392 +] +[ +-0.0140608 -1.16768 +-0.0140608 1.36283 +] +[ +-0.0070316 -1.19579 +-0.0070316 1.33471 +] +[ +-2.39929e-06 -1.25906 +-2.39929e-06 1.27145 +] +[ +0.0070268 -1.19579 +0.0070268 1.33471 +] +[ +0.014056 -1.16768 +0.014056 1.36283 +] +[ +0.0210852 -1.14659 +0.0210852 1.38392 +] +[ +0.0281144 -1.1255 +0.0281144 1.405 +] +[ +0.0351436 -1.11144 +0.0351436 1.41906 +] +[ +0.0421728 -1.09738 +0.0421728 1.43312 +] +[ +0.049202 -1.08333 +0.049202 1.44718 +] +[ +0.0562312 -1.06927 +0.0562312 1.46124 +] +[ +0.0632604 -1.06224 +0.0632604 1.46827 +] +[ +0.0702896 -1.04818 +0.0702896 1.48233 +] +[ +0.0773188 -1.04115 +0.0773188 1.48935 +] +[ +0.084348 -1.02709 +0.084348 1.50341 +] +[ +0.0913772 -1.02006 +0.0913772 1.51044 +] +[ +0.0984064 -1.00601 +0.0984064 1.51747 +] +[ +0.105436 -0.998976 +0.105436 1.5245 +] +[ +0.112465 -0.991947 +0.112465 1.53153 +] +[ +0.119494 -0.984918 +0.119494 1.54559 +] +[ +0.126523 -0.97086 +0.126523 1.55262 +] +[ +0.133552 -0.963831 +0.133552 1.55965 +] +[ +0.140582 -0.956801 +0.140582 1.56668 +] +[ +0.147611 -0.949772 +0.147611 1.5737 +] +[ +0.15464 -0.942743 +0.15464 1.58073 +] +[ +0.161669 -0.935714 +0.161669 1.58776 +] +[ +0.168698 -0.928685 +0.168698 1.58776 +] +[ +0.175728 -0.921656 +0.175728 1.59479 +] +[ +0.182757 -0.914626 +0.182757 1.60182 +] +[ +0.189786 -0.900568 +0.189786 1.60885 +] +[ +0.196815 -0.893539 +0.196815 1.61588 +] +[ +0.203844 -0.88651 +0.203844 1.62291 +] +[ +0.210874 -0.879481 +0.210874 1.62291 +] +[ +0.217903 -0.879481 +0.217903 1.62994 +] +[ +0.224932 -0.872451 +0.224932 1.63697 +] +[ +0.231961 -0.865422 +0.231961 1.644 +] +[ +0.23899 -0.858393 +0.23899 1.644 +] +[ +0.24602 -0.851364 +0.24602 1.65103 +] +[ +0.253049 -0.844335 +0.253049 1.65805 +] +[ +0.260078 -0.837305 +0.260078 1.65805 +] +[ +0.267107 -0.830276 +0.267107 1.66508 +] +[ +0.274137 -0.823247 +0.274137 1.67211 +] +[ +0.281166 -0.816218 +0.281166 1.67211 +] +[ +0.288195 -0.809189 +0.288195 1.67914 +] +[ +0.295224 -0.80216 +0.295224 1.67914 +] +[ +0.302253 -0.79513 +0.302253 1.68617 +] +[ +0.309283 -0.79513 +0.309283 1.6932 +] +[ +0.316312 -0.788101 +0.316312 1.6932 +] +[ +0.323341 -0.781072 +0.323341 1.70023 +] +[ +0.33037 -0.774043 +0.33037 1.70023 +] +[ +0.337399 -0.767014 +0.337399 1.70726 +] +[ +0.344429 -0.759985 +0.344429 1.70726 +] +[ +0.351458 -0.752955 +0.351458 1.71429 +] +[ +0.358487 -0.745926 +0.358487 1.71429 +] +[ +0.365516 -0.745926 +0.365516 1.72132 +] +[ +0.372545 -0.738897 +0.372545 1.72132 +] +[ +0.379575 -0.731868 +0.379575 1.72835 +] +[ +0.386604 -0.724839 +0.386604 1.72835 +] +[ +0.393633 -0.71781 +0.393633 1.73538 +] +[ +0.400662 -0.71078 +0.400662 1.73538 +] +[ +0.407691 -0.71078 +0.407691 1.73538 +] +[ +0.414721 -0.703751 +0.414721 1.7424 +] +[ +0.42175 -0.696722 +0.42175 1.7424 +] +[ +0.428779 -0.689693 +0.428779 1.74943 +] +[ +0.435808 -0.682664 +0.435808 1.74943 +] +[ +0.442837 -0.675635 +0.442837 1.74943 +] +[ +0.449867 -0.675635 +0.449867 1.75646 +] +[ +0.456896 -0.668605 +0.456896 1.75646 +] +[ +0.463925 -0.661576 +0.463925 1.75646 +] +[ +0.470954 -0.654547 +0.470954 1.76349 +] +[ +0.477983 -0.647518 +0.477983 1.76349 +] +[ +0.485013 -0.640489 +0.485013 1.76349 +] +[ +0.492042 -0.640489 +0.492042 1.77052 +] +[ +0.499071 -0.63346 +0.499071 1.77052 +] +[ +0.5061 -0.62643 +0.5061 1.77052 +] +[ +0.513129 -0.619401 +0.513129 1.77755 +] +[ +0.520159 -0.612372 +0.520159 1.77755 +] +[ +0.527188 -0.605343 +0.527188 1.77755 +] +[ +0.534217 -0.605343 +0.534217 1.77755 +] +[ +0.541246 -0.598314 +0.541246 1.78458 +] +[ +0.548275 -0.591285 +0.548275 1.78458 +] +[ +0.555305 -0.584255 +0.555305 1.78458 +] +[ +0.562334 -0.577226 +0.562334 1.78458 +] +[ +0.569363 -0.570197 +0.569363 1.79161 +] +[ +0.576392 -0.563168 +0.576392 1.79161 +] +[ +0.583421 -0.563168 +0.583421 1.79161 +] +[ +0.59045 -0.556139 +0.59045 1.79161 +] +[ +0.59748 -0.549109 +0.59748 1.79161 +] +[ +0.604509 -0.54208 +0.604509 1.79864 +] +[ +0.611538 -0.535051 +0.611538 1.79864 +] +[ +0.618567 -0.528022 +0.618567 1.79864 +] +[ +0.625596 -0.528022 +0.625596 1.79864 +] +[ +0.632625 -0.520993 +0.632625 1.79864 +] +[ +0.639655 -0.513964 +0.639655 1.79864 +] +[ +0.646684 -0.506934 +0.646684 1.80567 +] +[ +0.653713 -0.499905 +0.653713 1.80567 +] +[ +0.660742 -0.492876 +0.660742 1.80567 +] +[ +0.667771 -0.485847 +0.667771 1.80567 +] +[ +0.6748 -0.485847 +0.6748 1.80567 +] +[ +0.68183 -0.478818 +0.68183 1.80567 +] +[ +0.688859 -0.471788 +0.688859 1.80567 +] +[ +0.695888 -0.464759 +0.695888 1.80567 +] +[ +0.702917 -0.45773 +0.702917 1.80567 +] +[ +0.709946 -0.450701 +0.709946 1.80567 +] +[ +0.716976 -0.443672 +0.716976 1.8127 +] +[ +0.724005 -0.436642 +0.724005 1.8127 +] +[ +0.731034 -0.436642 +0.731034 1.8127 +] +[ +0.738063 -0.429613 +0.738063 1.8127 +] +[ +0.745092 -0.422584 +0.745092 1.8127 +] +[ +0.752121 -0.415555 +0.752121 1.8127 +] +[ +0.759151 -0.408526 +0.759151 1.8127 +] +[ +0.76618 -0.401496 +0.76618 1.8127 +] +[ +0.773209 -0.394467 +0.773209 1.8127 +] +[ +0.780238 -0.387438 +0.780238 1.8127 +] +[ +0.787267 -0.380409 +0.787267 1.8127 +] +[ +0.794296 -0.37338 +0.794296 1.8127 +] +[ +0.801326 -0.36635 +0.801326 1.8127 +] +[ +0.808355 -0.36635 +0.808355 1.8127 +] +[ +0.815384 -0.359321 +0.815384 1.8127 +] +[ +0.822413 -0.352292 +0.822413 1.8127 +] +[ +0.829442 -0.345263 +0.829442 1.8127 +] +[ +0.836472 -0.338234 +0.836472 1.8127 +] +[ +0.843501 -0.331204 +0.843501 1.80567 +] +[ +0.85053 -0.324175 +0.85053 1.80567 +] +[ +0.857559 -0.317146 +0.857559 1.80567 +] +[ +0.864588 -0.310117 +0.864588 1.80567 +] +[ +0.871617 -0.303088 +0.871617 1.80567 +] +[ +0.878647 -0.296058 +0.878647 1.80567 +] +[ +0.885676 -0.289029 +0.885676 1.80567 +] +[ +0.892705 -0.282 +0.892705 1.80567 +] +[ +0.899734 -0.274971 +0.899734 1.80567 +] +[ +0.906763 -0.267942 +0.906763 1.80567 +] +[ +0.913792 -0.260912 +0.913792 1.79864 +] +[ +0.920822 -0.253883 +0.920822 1.79864 +] +[ +0.927851 -0.246854 +0.927851 1.79864 +] +[ +0.93488 -0.239825 +0.93488 1.79864 +] +[ +0.941909 -0.232796 +0.941909 1.79864 +] +[ +0.948938 -0.225766 +0.948938 1.79864 +] +[ +0.955967 -0.218737 +0.955967 1.79161 +] +[ +0.962997 -0.211708 +0.962997 1.79161 +] +[ +0.970026 -0.204679 +0.970026 1.79161 +] +[ +0.977055 -0.19765 +0.977055 1.79161 +] +[ +0.984084 -0.19062 +0.984084 1.78458 +] +[ +0.991113 -0.183591 +0.991113 1.78458 +] +[ +0.998143 -0.176562 +0.998143 1.78458 +] +[ +1.00517 -0.169533 +1.00517 1.78458 +] +[ +1.0122 -0.162504 +1.0122 1.77755 +] +[ +1.01923 -0.155474 +1.01923 1.77755 +] +[ +1.02626 -0.148445 +1.02626 1.77755 +] +[ +1.03329 -0.141416 +1.03329 1.77755 +] +[ +1.04032 -0.134387 +1.04032 1.77052 +] +[ +1.04735 -0.120328 +1.04735 1.77052 +] +[ +1.05438 -0.113299 +1.05438 1.77052 +] +[ +1.06141 -0.10627 +1.06141 1.76349 +] +[ +1.06843 -0.0992407 +1.06843 1.76349 +] +[ +1.07546 -0.0922115 +1.07546 1.76349 +] +[ +1.08249 -0.0851823 +1.08249 1.75646 +] +[ +1.08952 -0.0781531 +1.08952 1.75646 +] +[ +1.09655 -0.0711239 +1.09655 1.74943 +] +[ +1.10358 -0.0570655 +1.10358 1.74943 +] +[ +1.11061 -0.0500363 +1.11061 1.74943 +] +[ +1.11764 -0.0430071 +1.11764 1.7424 +] +[ +1.12467 -0.0359779 +1.12467 1.7424 +] +[ +1.1317 -0.0289487 +1.1317 1.73538 +] +[ +1.13873 -0.0148903 +1.13873 1.73538 +] +[ +1.14576 -0.0078611 +1.14576 1.72835 +] +[ +1.15279 -0.000831896 +1.15279 1.72835 +] +[ +1.15981 0.0061973 +1.15981 1.72132 +] +[ +1.16684 0.0202557 +1.16684 1.72132 +] +[ +1.17387 0.0272849 +1.17387 1.71429 +] +[ +1.1809 0.0343141 +1.1809 1.71429 +] +[ +1.18793 0.0413433 +1.18793 1.70726 +] +[ +1.19496 0.0554017 +1.19496 1.70726 +] +[ +1.20199 0.0624309 +1.20199 1.70023 +] +[ +1.20902 0.0694601 +1.20902 1.70023 +] +[ +1.21605 0.0835185 +1.21605 1.6932 +] +[ +1.22308 0.0905477 +1.22308 1.68617 +] +[ +1.23011 0.0975769 +1.23011 1.68617 +] +[ +1.23714 0.111635 +1.23714 1.67914 +] +[ +1.24417 0.118665 +1.24417 1.67914 +] +[ +1.25119 0.125694 +1.25119 1.67211 +] +[ +1.25822 0.139752 +1.25822 1.66508 +] +[ +1.26525 0.146781 +1.26525 1.65805 +] +[ +1.27228 0.16084 +1.27228 1.65805 +] +[ +1.27931 0.167869 +1.27931 1.65103 +] +[ +1.28634 0.181927 +1.28634 1.644 +] +[ +1.29337 0.188957 +1.29337 1.63697 +] +[ +1.3004 0.203015 +1.3004 1.63697 +] +[ +1.30743 0.210044 +1.30743 1.62994 +] +[ +1.31446 0.224103 +1.31446 1.62291 +] +[ +1.32149 0.231132 +1.32149 1.61588 +] +[ +1.32852 0.24519 +1.32852 1.60885 +] +[ +1.33555 0.252219 +1.33555 1.60182 +] +[ +1.34258 0.266278 +1.34258 1.59479 +] +[ +1.3496 0.280336 +1.3496 1.58776 +] +[ +1.35663 0.287365 +1.35663 1.58073 +] +[ +1.36366 0.301424 +1.36366 1.5737 +] +[ +1.37069 0.315482 +1.37069 1.56668 +] +[ +1.37772 0.329541 +1.37772 1.55965 +] +[ +1.38475 0.33657 +1.38475 1.55262 +] +[ +1.39178 0.350628 +1.39178 1.54559 +] +[ +1.39881 0.364687 +1.39881 1.53856 +] +[ +1.40584 0.378745 +1.40584 1.53153 +] +[ +1.41287 0.392803 +1.41287 1.51747 +] +[ +1.4199 0.406862 +1.4199 1.51044 +] +[ +1.42693 0.42092 +1.42693 1.50341 +] +[ +1.43396 0.434979 +1.43396 1.48935 +] +[ +1.44098 0.449037 +1.44098 1.48233 +] +[ +1.44801 0.463095 +1.44801 1.46827 +] +[ +1.45504 0.477154 +1.45504 1.46124 +] +[ +1.46207 0.498241 +1.46207 1.44718 +] +[ +1.4691 0.5123 +1.4691 1.44015 +] +[ +1.47613 0.526358 +1.47613 1.42609 +] +[ +1.48316 0.547446 +1.48316 1.41203 +] +[ +1.49019 0.561504 +1.49019 1.39798 +] +[ +1.49722 0.582591 +1.49722 1.38392 +] +[ +1.50425 0.603679 +1.50425 1.36986 +] +[ +1.51128 0.624766 +1.51128 1.3558 +] +[ +1.51831 0.645854 +1.51831 1.33471 +] +[ +1.52534 0.666941 +1.52534 1.32065 +] +[ +1.53236 0.695058 +1.53236 1.29957 +] +[ +1.53939 0.716146 +1.53939 1.27848 +] +[ +1.54642 0.744262 +1.54642 1.25036 +] +[ +1.55345 0.779408 +1.55345 1.22225 +] +[ +1.56048 0.814554 +1.56048 1.19413 +] +[ +1.56751 0.856729 +1.56751 1.15195 +] +[ +1.57454 0.919992 +1.57454 1.09572 +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +-2.39929e-06 -1.25906 +0.0070268 -1.25906 +] +[ +-2.39929e-06 -1.25203 +0.0070268 -1.25203 +] +[ +-2.39929e-06 -1.245 +0.0070268 -1.245 +] +[ +-2.39929e-06 -1.23797 +0.0070268 -1.23797 +] +[ +-2.39929e-06 -1.23094 +0.0070268 -1.23094 +] +[ +-2.39929e-06 -1.22391 +0.0070268 -1.22391 +] +[ +-2.39929e-06 -1.21688 +0.0070268 -1.21688 +] +[ +-2.39929e-06 -1.20985 +0.0070268 -1.20985 +] +[ +-2.39929e-06 -1.20282 +0.0070268 -1.20282 +] +[ +-0.0070316 -1.19579 +0.014056 -1.19579 +] +[ +-0.0070316 -1.18876 +0.014056 -1.18876 +] +[ +-0.0070316 -1.18173 +0.014056 -1.18173 +] +[ +-0.0070316 -1.17471 +0.014056 -1.17471 +] +[ +-0.0140608 -1.16768 +0.0210852 -1.16768 +] +[ +-0.0140608 -1.16065 +0.0210852 -1.16065 +] +[ +-0.0140608 -1.15362 +0.0210852 -1.15362 +] +[ +-0.02109 -1.14659 +0.0281144 -1.14659 +] +[ +-0.02109 -1.13956 +0.0281144 -1.13956 +] +[ +-0.02109 -1.13253 +0.0281144 -1.13253 +] +[ +-0.0281192 -1.1255 +0.0351436 -1.1255 +] +[ +-0.0281192 -1.11847 +0.0351436 -1.11847 +] +[ +-0.0351484 -1.11144 +0.0421728 -1.11144 +] +[ +-0.0351484 -1.10441 +0.0421728 -1.10441 +] +[ +-0.0421776 -1.09738 +0.049202 -1.09738 +] +[ +-0.0421776 -1.09036 +0.049202 -1.09036 +] +[ +-0.0492068 -1.08333 +0.0562312 -1.08333 +] +[ +-0.0492068 -1.0763 +0.0562312 -1.0763 +] +[ +-0.056236 -1.06927 +0.0632604 -1.06927 +] +[ +-0.0632652 -1.06224 +0.0702896 -1.06224 +] +[ +-0.0632652 -1.05521 +0.0702896 -1.05521 +] +[ +-0.0702944 -1.04818 +0.0773188 -1.04818 +] +[ +-0.0773236 -1.04115 +0.084348 -1.04115 +] +[ +-0.0773236 -1.03412 +0.084348 -1.03412 +] +[ +-0.0843528 -1.02709 +0.0913772 -1.02709 +] +[ +-0.091382 -1.02006 +0.0984064 -1.02006 +] +[ +-0.091382 -1.01303 +0.0984064 -1.01303 +] +[ +-0.0984112 -1.00601 +0.105436 -1.00601 +] +[ +-0.10544 -0.998976 +0.112465 -0.998976 +] +[ +-0.11247 -0.991947 +0.119494 -0.991947 +] +[ +-0.119499 -0.984918 +0.126523 -0.984918 +] +[ +-0.119499 -0.977889 +0.126523 -0.977889 +] +[ +-0.126528 -0.97086 +0.133552 -0.97086 +] +[ +-0.133557 -0.963831 +0.140582 -0.963831 +] +[ +-0.140586 -0.956801 +0.147611 -0.956801 +] +[ +-0.147616 -0.949772 +0.15464 -0.949772 +] +[ +-0.154645 -0.942743 +0.161669 -0.942743 +] +[ +-0.161674 -0.935714 +0.168698 -0.935714 +] +[ +-0.168703 -0.928685 +0.175728 -0.928685 +] +[ +-0.175732 -0.921656 +0.182757 -0.921656 +] +[ +-0.182762 -0.914626 +0.189786 -0.914626 +] +[ +-0.182762 -0.907597 +0.189786 -0.907597 +] +[ +-0.189791 -0.900568 +0.196815 -0.900568 +] +[ +-0.19682 -0.893539 +0.203844 -0.893539 +] +[ +-0.203849 -0.88651 +0.210874 -0.88651 +] +[ +-0.217908 -0.879481 +0.224932 -0.879481 +] +[ +-0.224937 -0.872451 +0.231961 -0.872451 +] +[ +-0.231966 -0.865422 +0.23899 -0.865422 +] +[ +-0.238995 -0.858393 +0.24602 -0.858393 +] +[ +-0.246024 -0.851364 +0.253049 -0.851364 +] +[ +-0.253054 -0.844335 +0.260078 -0.844335 +] +[ +-0.260083 -0.837305 +0.267107 -0.837305 +] +[ +-0.267112 -0.830276 +0.274137 -0.830276 +] +[ +-0.274141 -0.823247 +0.281166 -0.823247 +] +[ +-0.281171 -0.816218 +0.288195 -0.816218 +] +[ +-0.2882 -0.809189 +0.295224 -0.809189 +] +[ +-0.295229 -0.80216 +0.302253 -0.80216 +] +[ +-0.309287 -0.79513 +0.316312 -0.79513 +] +[ +-0.316317 -0.788101 +0.323341 -0.788101 +] +[ +-0.323346 -0.781072 +0.33037 -0.781072 +] +[ +-0.330375 -0.774043 +0.337399 -0.774043 +] +[ +-0.337404 -0.767014 +0.344429 -0.767014 +] +[ +-0.344433 -0.759985 +0.351458 -0.759985 +] +[ +-0.351463 -0.752955 +0.358487 -0.752955 +] +[ +-0.365521 -0.745926 +0.372545 -0.745926 +] +[ +-0.37255 -0.738897 +0.379575 -0.738897 +] +[ +-0.379579 -0.731868 +0.386604 -0.731868 +] +[ +-0.386609 -0.724839 +0.393633 -0.724839 +] +[ +-0.393638 -0.71781 +0.400662 -0.71781 +] +[ +-0.407696 -0.71078 +0.414721 -0.71078 +] +[ +-0.414725 -0.703751 +0.42175 -0.703751 +] +[ +-0.421755 -0.696722 +0.428779 -0.696722 +] +[ +-0.428784 -0.689693 +0.435808 -0.689693 +] +[ +-0.435813 -0.682664 +0.442837 -0.682664 +] +[ +-0.449871 -0.675635 +0.456896 -0.675635 +] +[ +-0.456901 -0.668605 +0.463925 -0.668605 +] +[ +-0.46393 -0.661576 +0.470954 -0.661576 +] +[ +-0.470959 -0.654547 +0.477983 -0.654547 +] +[ +-0.477988 -0.647518 +0.485013 -0.647518 +] +[ +-0.492047 -0.640489 +0.499071 -0.640489 +] +[ +-0.499076 -0.63346 +0.5061 -0.63346 +] +[ +-0.506105 -0.62643 +0.513129 -0.62643 +] +[ +-0.513134 -0.619401 +0.520159 -0.619401 +] +[ +-0.520163 -0.612372 +0.527188 -0.612372 +] +[ +-0.534222 -0.605343 +0.541246 -0.605343 +] +[ +-0.541251 -0.598314 +0.548275 -0.598314 +] +[ +-0.54828 -0.591285 +0.555305 -0.591285 +] +[ +-0.555309 -0.584255 +0.562334 -0.584255 +] +[ +-0.562338 -0.577226 +0.569363 -0.577226 +] +[ +-0.569368 -0.570197 +0.576392 -0.570197 +] +[ +-0.583426 -0.563168 +0.59045 -0.563168 +] +[ +-0.590455 -0.556139 +0.59748 -0.556139 +] +[ +-0.597484 -0.549109 +0.604509 -0.549109 +] +[ +-0.604514 -0.54208 +0.611538 -0.54208 +] +[ +-0.611543 -0.535051 +0.618567 -0.535051 +] +[ +-0.625601 -0.528022 +0.632625 -0.528022 +] +[ +-0.63263 -0.520993 +0.639655 -0.520993 +] +[ +-0.639659 -0.513964 +0.646684 -0.513964 +] +[ +-0.646689 -0.506934 +0.653713 -0.506934 +] +[ +-0.653718 -0.499905 +0.660742 -0.499905 +] +[ +-0.660747 -0.492876 +0.667771 -0.492876 +] +[ +-0.674805 -0.485847 +0.68183 -0.485847 +] +[ +-0.681834 -0.478818 +0.688859 -0.478818 +] +[ +-0.688864 -0.471788 +0.695888 -0.471788 +] +[ +-0.695893 -0.464759 +0.702917 -0.464759 +] +[ +-0.702922 -0.45773 +0.709946 -0.45773 +] +[ +-0.709951 -0.450701 +0.716976 -0.450701 +] +[ +-0.71698 -0.443672 +0.724005 -0.443672 +] +[ +-0.731039 -0.436642 +0.738063 -0.436642 +] +[ +-0.738068 -0.429613 +0.745092 -0.429613 +] +[ +-0.745097 -0.422584 +0.752121 -0.422584 +] +[ +-0.752126 -0.415555 +0.759151 -0.415555 +] +[ +-0.759155 -0.408526 +0.76618 -0.408526 +] +[ +-0.766185 -0.401496 +0.773209 -0.401496 +] +[ +-0.773214 -0.394467 +0.780238 -0.394467 +] +[ +-0.780243 -0.387438 +0.787267 -0.387438 +] +[ +-0.787272 -0.380409 +0.794296 -0.380409 +] +[ +-0.794301 -0.37338 +0.801326 -0.37338 +] +[ +-0.80836 -0.36635 +0.815384 -0.36635 +] +[ +-0.815389 -0.359321 +0.822413 -0.359321 +] +[ +-0.822418 -0.352292 +0.829442 -0.352292 +] +[ +-0.829447 -0.345263 +0.836472 -0.345263 +] +[ +-0.836476 -0.338234 +0.843501 -0.338234 +] +[ +-0.843506 -0.331204 +0.85053 -0.331204 +] +[ +-0.850535 -0.324175 +0.857559 -0.324175 +] +[ +-0.857564 -0.317146 +0.864588 -0.317146 +] +[ +-0.864593 -0.310117 +0.871617 -0.310117 +] +[ +-0.871622 -0.303088 +0.878647 -0.303088 +] +[ +-0.878651 -0.296058 +0.885676 -0.296058 +] +[ +-0.88568 -0.289029 +0.892705 -0.289029 +] +[ +-0.89271 -0.282 +0.899734 -0.282 +] +[ +-0.899739 -0.274971 +0.906763 -0.274971 +] +[ +-0.906768 -0.267942 +0.913792 -0.267942 +] +[ +-0.913797 -0.260912 +0.920822 -0.260912 +] +[ +-0.920826 -0.253883 +0.927851 -0.253883 +] +[ +-0.927856 -0.246854 +0.93488 -0.246854 +] +[ +-0.934885 -0.239825 +0.941909 -0.239825 +] +[ +-0.941914 -0.232796 +0.948938 -0.232796 +] +[ +-0.948943 -0.225766 +0.955967 -0.225766 +] +[ +-0.955972 -0.218737 +0.962997 -0.218737 +] +[ +-0.963001 -0.211708 +0.970026 -0.211708 +] +[ +-0.970031 -0.204679 +0.977055 -0.204679 +] +[ +-0.97706 -0.19765 +0.984084 -0.19765 +] +[ +-0.984089 -0.19062 +0.991113 -0.19062 +] +[ +-0.991118 -0.183591 +0.998143 -0.183591 +] +[ +-0.998147 -0.176562 +1.00517 -0.176562 +] +[ +-1.00518 -0.169533 +1.0122 -0.169533 +] +[ +-1.01221 -0.162504 +1.01923 -0.162504 +] +[ +-1.01924 -0.155474 +1.02626 -0.155474 +] +[ +-1.02626 -0.148445 +1.03329 -0.148445 +] +[ +-1.03329 -0.141416 +1.04032 -0.141416 +] +[ +-1.04032 -0.134387 +1.04735 -0.134387 +] +[ +-1.04032 -0.127358 +1.04735 -0.127358 +] +[ +-1.04735 -0.120328 +1.05438 -0.120328 +] +[ +-1.05438 -0.113299 +1.06141 -0.113299 +] +[ +-1.06141 -0.10627 +1.06843 -0.10627 +] +[ +-1.06844 -0.0992407 +1.07546 -0.0992407 +] +[ +-1.07547 -0.0922115 +1.08249 -0.0922115 +] +[ +-1.0825 -0.0851823 +1.08952 -0.0851823 +] +[ +-1.08953 -0.0781531 +1.09655 -0.0781531 +] +[ +-1.09656 -0.0711239 +1.10358 -0.0711239 +] +[ +-1.09656 -0.0640947 +1.10358 -0.0640947 +] +[ +-1.10359 -0.0570655 +1.11061 -0.0570655 +] +[ +-1.11062 -0.0500363 +1.11764 -0.0500363 +] +[ +-1.11764 -0.0430071 +1.12467 -0.0430071 +] +[ +-1.12467 -0.0359779 +1.1317 -0.0359779 +] +[ +-1.1317 -0.0289487 +1.13873 -0.0289487 +] +[ +-1.1317 -0.0219195 +1.13873 -0.0219195 +] +[ +-1.13873 -0.0148903 +1.14576 -0.0148903 +] +[ +-1.14576 -0.0078611 +1.15279 -0.0078611 +] +[ +-1.15279 -0.000831896 +1.15981 -0.000831896 +] +[ +-1.15982 0.0061973 +1.16684 0.0061973 +] +[ +-1.15982 0.0132265 +1.16684 0.0132265 +] +[ +-1.16685 0.0202557 +1.17387 0.0202557 +] +[ +-1.17388 0.0272849 +1.1809 0.0272849 +] +[ +-1.18091 0.0343141 +1.18793 0.0343141 +] +[ +-1.18794 0.0413433 +1.19496 0.0413433 +] +[ +-1.18794 0.0483725 +1.19496 0.0483725 +] +[ +-1.19497 0.0554017 +1.20199 0.0554017 +] +[ +-1.202 0.0624309 +1.20902 0.0624309 +] +[ +-1.20902 0.0694601 +1.21605 0.0694601 +] +[ +-1.20902 0.0764893 +1.21605 0.0764893 +] +[ +-1.21605 0.0835185 +1.22308 0.0835185 +] +[ +-1.22308 0.0905477 +1.23011 0.0905477 +] +[ +-1.23011 0.0975769 +1.23714 0.0975769 +] +[ +-1.23011 0.104606 +1.23714 0.104606 +] +[ +-1.23714 0.111635 +1.24417 0.111635 +] +[ +-1.24417 0.118665 +1.25119 0.118665 +] +[ +-1.2512 0.125694 +1.25822 0.125694 +] +[ +-1.2512 0.132723 +1.25822 0.132723 +] +[ +-1.25823 0.139752 +1.26525 0.139752 +] +[ +-1.26526 0.146781 +1.27228 0.146781 +] +[ +-1.26526 0.153811 +1.27228 0.153811 +] +[ +-1.27229 0.16084 +1.27931 0.16084 +] +[ +-1.27932 0.167869 +1.28634 0.167869 +] +[ +-1.27932 0.174898 +1.28634 0.174898 +] +[ +-1.28635 0.181927 +1.29337 0.181927 +] +[ +-1.29338 0.188957 +1.3004 0.188957 +] +[ +-1.29338 0.195986 +1.3004 0.195986 +] +[ +-1.3004 0.203015 +1.30743 0.203015 +] +[ +-1.30743 0.210044 +1.31446 0.210044 +] +[ +-1.30743 0.217073 +1.31446 0.217073 +] +[ +-1.31446 0.224103 +1.32149 0.224103 +] +[ +-1.32149 0.231132 +1.32852 0.231132 +] +[ +-1.32149 0.238161 +1.32852 0.238161 +] +[ +-1.32852 0.24519 +1.33555 0.24519 +] +[ +-1.33555 0.252219 +1.34258 0.252219 +] +[ +-1.33555 0.259249 +1.34258 0.259249 +] +[ +-1.34258 0.266278 +1.3496 0.266278 +] +[ +-1.34258 0.273307 +1.3496 0.273307 +] +[ +-1.34961 0.280336 +1.35663 0.280336 +] +[ +-1.35664 0.287365 +1.36366 0.287365 +] +[ +-1.35664 0.294395 +1.36366 0.294395 +] +[ +-1.36367 0.301424 +1.37069 0.301424 +] +[ +-1.36367 0.308453 +1.37069 0.308453 +] +[ +-1.3707 0.315482 +1.37772 0.315482 +] +[ +-1.3707 0.322511 +1.37772 0.322511 +] +[ +-1.37773 0.329541 +1.38475 0.329541 +] +[ +-1.38476 0.33657 +1.39178 0.33657 +] +[ +-1.38476 0.343599 +1.39178 0.343599 +] +[ +-1.39178 0.350628 +1.39881 0.350628 +] +[ +-1.39178 0.357657 +1.39881 0.357657 +] +[ +-1.39881 0.364687 +1.40584 0.364687 +] +[ +-1.39881 0.371716 +1.40584 0.371716 +] +[ +-1.40584 0.378745 +1.41287 0.378745 +] +[ +-1.40584 0.385774 +1.41287 0.385774 +] +[ +-1.41287 0.392803 +1.4199 0.392803 +] +[ +-1.41287 0.399833 +1.4199 0.399833 +] +[ +-1.4199 0.406862 +1.42693 0.406862 +] +[ +-1.4199 0.413891 +1.42693 0.413891 +] +[ +-1.42693 0.42092 +1.43396 0.42092 +] +[ +-1.42693 0.427949 +1.43396 0.427949 +] +[ +-1.43396 0.434979 +1.44098 0.434979 +] +[ +-1.43396 0.442008 +1.44098 0.442008 +] +[ +-1.44099 0.449037 +1.44801 0.449037 +] +[ +-1.44099 0.456066 +1.44801 0.456066 +] +[ +-1.44802 0.463095 +1.45504 0.463095 +] +[ +-1.44802 0.470125 +1.45504 0.470125 +] +[ +-1.45505 0.477154 +1.46207 0.477154 +] +[ +-1.45505 0.484183 +1.46207 0.484183 +] +[ +-1.45505 0.491212 +1.46207 0.491212 +] +[ +-1.46208 0.498241 +1.4691 0.498241 +] +[ +-1.46208 0.505271 +1.4691 0.505271 +] +[ +-1.46911 0.5123 +1.47613 0.5123 +] +[ +-1.46911 0.519329 +1.47613 0.519329 +] +[ +-1.47614 0.526358 +1.48316 0.526358 +] +[ +-1.47614 0.533387 +1.48316 0.533387 +] +[ +-1.47614 0.540416 +1.48316 0.540416 +] +[ +-1.48316 0.547446 +1.49019 0.547446 +] +[ +-1.48316 0.554475 +1.49019 0.554475 +] +[ +-1.49019 0.561504 +1.49722 0.561504 +] +[ +-1.49019 0.568533 +1.49722 0.568533 +] +[ +-1.49019 0.575562 +1.49722 0.575562 +] +[ +-1.49722 0.582591 +1.50425 0.582591 +] +[ +-1.49722 0.589621 +1.50425 0.589621 +] +[ +-1.49722 0.59665 +1.50425 0.59665 +] +[ +-1.50425 0.603679 +1.51128 0.603679 +] +[ +-1.50425 0.610708 +1.51128 0.610708 +] +[ +-1.50425 0.617737 +1.51128 0.617737 +] +[ +-1.51128 0.624766 +1.51831 0.624766 +] +[ +-1.51128 0.631796 +1.51831 0.631796 +] +[ +-1.51128 0.638825 +1.51831 0.638825 +] +[ +-1.51831 0.645854 +1.52534 0.645854 +] +[ +-1.51831 0.652883 +1.52534 0.652883 +] +[ +-1.51831 0.659912 +1.52534 0.659912 +] +[ +-1.52534 0.666941 +1.53236 0.666941 +] +[ +-1.52534 0.673971 +1.53236 0.673971 +] +[ +-1.52534 0.681 +1.53236 0.681 +] +[ +-1.52534 0.688029 +1.53236 0.688029 +] +[ +-1.53237 0.695058 +1.53939 0.695058 +] +[ +-1.53237 0.702087 +1.53939 0.702087 +] +[ +-1.53237 0.709117 +1.53939 0.709117 +] +[ +-1.5394 0.716146 +1.54642 0.716146 +] +[ +-1.5394 0.723175 +1.54642 0.723175 +] +[ +-1.5394 0.730204 +1.54642 0.730204 +] +[ +-1.5394 0.737233 +1.54642 0.737233 +] +[ +-1.54643 0.744262 +1.55345 0.744262 +] +[ +-1.54643 0.751292 +1.55345 0.751292 +] +[ +-1.54643 0.758321 +1.55345 0.758321 +] +[ +-1.54643 0.76535 +1.55345 0.76535 +] +[ +-1.54643 0.772379 +1.55345 0.772379 +] +[ +-1.55346 0.779408 +1.56048 0.779408 +] +[ +-1.55346 0.786437 +1.56048 0.786437 +] +[ +-1.55346 0.793467 +1.56048 0.793467 +] +[ +-1.55346 0.800496 +1.56048 0.800496 +] +[ +-1.55346 0.807525 +1.56048 0.807525 +] +[ +-1.56049 0.814554 +1.56751 0.814554 +] +[ +-1.56049 0.821583 +1.56751 0.821583 +] +[ +-1.56049 0.828612 +1.56751 0.828612 +] +[ +-1.56049 0.835642 +1.56751 0.835642 +] +[ +-1.56049 0.842671 +1.56751 0.842671 +] +[ +-1.56049 0.8497 +1.56751 0.8497 +] +[ +-1.56752 0.856729 +1.57454 0.856729 +] +[ +-1.56752 0.863758 +1.57454 0.863758 +] +[ +-1.56752 0.870788 +1.57454 0.870788 +] +[ +-1.56752 0.877817 +1.57454 0.877817 +] +[ +-1.56752 0.884846 +1.57454 0.884846 +] +[ +-1.56752 0.891875 +1.57454 0.891875 +] +[ +-1.56752 0.898904 +1.57454 0.898904 +] +[ +-1.56752 0.905933 +1.57454 0.905933 +] +[ +-1.56752 0.912963 +1.57454 0.912963 +] +[ +-1.57454 0.919992 +1.58157 0.919992 +] +[ +-1.57454 0.927021 +1.58157 0.927021 +] +[ +-1.57454 0.93405 +1.58157 0.93405 +] +[ +-1.57454 0.941079 +1.58157 0.941079 +] +[ +-1.57454 0.948108 +1.58157 0.948108 +] +[ +-1.57454 0.955138 +1.58157 0.955138 +] +[ +-1.57454 0.962167 +1.58157 0.962167 +] +[ +-1.57454 0.969196 +1.58157 0.969196 +] +[ +-1.57454 0.976225 +1.58157 0.976225 +] +[ +-1.57454 0.983254 +1.58157 0.983254 +] +[ +-1.57454 0.990283 +1.58157 0.990283 +] +[ +-1.57454 0.997313 +1.58157 0.997313 +] +[ +-1.57454 1.00434 +1.58157 1.00434 +] +[ +-1.57454 1.01137 +1.58157 1.01137 +] +[ +-1.57454 1.0184 +1.58157 1.0184 +] +[ +-1.57454 1.02543 +1.58157 1.02543 +] +[ +-1.57454 1.03246 +1.58157 1.03246 +] +[ +-1.57454 1.03949 +1.58157 1.03949 +] +[ +-1.57454 1.04652 +1.58157 1.04652 +] +[ +-1.57454 1.05355 +1.58157 1.05355 +] +[ +-1.57454 1.06058 +1.58157 1.06058 +] +[ +-1.57454 1.0676 +1.58157 1.0676 +] +[ +-1.57454 1.07463 +1.58157 1.07463 +] +[ +-1.57454 1.08166 +1.58157 1.08166 +] +[ +-1.57454 1.08869 +1.58157 1.08869 +] +[ +-1.56752 1.09572 +1.57454 1.09572 +] +[ +-1.56752 1.10275 +1.57454 1.10275 +] +[ +-1.56752 1.10978 +1.57454 1.10978 +] +[ +-1.56752 1.11681 +1.57454 1.11681 +] +[ +-1.56752 1.12384 +1.57454 1.12384 +] +[ +-1.56752 1.13087 +1.57454 1.13087 +] +[ +-1.56752 1.1379 +1.57454 1.1379 +] +[ +-1.56752 1.14493 +1.57454 1.14493 +] +[ +-1.56049 1.15195 +1.56751 1.15195 +] +[ +-1.56049 1.15898 +1.56751 1.15898 +] +[ +-1.56049 1.16601 +1.56751 1.16601 +] +[ +-1.56049 1.17304 +1.56751 1.17304 +] +[ +-1.56049 1.18007 +1.56751 1.18007 +] +[ +-1.56049 1.1871 +1.56751 1.1871 +] +[ +-1.55346 1.19413 +1.56048 1.19413 +] +[ +-1.55346 1.20116 +1.56048 1.20116 +] +[ +-1.55346 1.20819 +1.56048 1.20819 +] +[ +-1.55346 1.21522 +1.56048 1.21522 +] +[ +-1.54643 1.22225 +1.55345 1.22225 +] +[ +-1.54643 1.22928 +1.55345 1.22928 +] +[ +-1.54643 1.2363 +1.55345 1.2363 +] +[ +-1.54643 1.24333 +1.55345 1.24333 +] +[ +-1.5394 1.25036 +1.54642 1.25036 +] +[ +-1.5394 1.25739 +1.54642 1.25739 +] +[ +-1.5394 1.26442 +1.54642 1.26442 +] +[ +-1.5394 1.27145 +-2.39929e-06 1.27145 +0.0070268 1.27145 +1.54642 1.27145 +] +[ +-1.53237 1.27848 +-2.39929e-06 1.27848 +0.0070268 1.27848 +1.53939 1.27848 +] +[ +-1.53237 1.28551 +-2.39929e-06 1.28551 +0.0070268 1.28551 +1.53939 1.28551 +] +[ +-1.53237 1.29254 +-2.39929e-06 1.29254 +0.0070268 1.29254 +1.53939 1.29254 +] +[ +-1.52534 1.29957 +-2.39929e-06 1.29957 +0.0070268 1.29957 +1.53236 1.29957 +] +[ +-1.52534 1.3066 +-2.39929e-06 1.3066 +0.0070268 1.3066 +1.53236 1.3066 +] +[ +-1.52534 1.31363 +-2.39929e-06 1.31363 +0.0070268 1.31363 +1.53236 1.31363 +] +[ +-1.51831 1.32065 +-2.39929e-06 1.32065 +0.0070268 1.32065 +1.52534 1.32065 +] +[ +-1.51831 1.32768 +-2.39929e-06 1.32768 +0.0070268 1.32768 +1.52534 1.32768 +] +[ +-1.51128 1.33471 +-0.0070316 1.33471 +0.014056 1.33471 +1.51831 1.33471 +] +[ +-1.51128 1.34174 +-0.0070316 1.34174 +0.014056 1.34174 +1.51831 1.34174 +] +[ +-1.51128 1.34877 +-0.0070316 1.34877 +0.014056 1.34877 +1.51831 1.34877 +] +[ +-1.50425 1.3558 +-0.0070316 1.3558 +0.014056 1.3558 +1.51128 1.3558 +] +[ +-1.50425 1.36283 +-0.0140608 1.36283 +0.0210852 1.36283 +1.51128 1.36283 +] +[ +-1.49722 1.36986 +-0.0140608 1.36986 +0.0210852 1.36986 +1.50425 1.36986 +] +[ +-1.49722 1.37689 +-0.0140608 1.37689 +0.0210852 1.37689 +1.50425 1.37689 +] +[ +-1.49019 1.38392 +-0.02109 1.38392 +0.0281144 1.38392 +1.49722 1.38392 +] +[ +-1.49019 1.39095 +-0.02109 1.39095 +0.0281144 1.39095 +1.49722 1.39095 +] +[ +-1.48316 1.39798 +-0.02109 1.39798 +0.0281144 1.39798 +1.49019 1.39798 +] +[ +-1.48316 1.405 +-0.0281192 1.405 +0.0351436 1.405 +1.49019 1.405 +] +[ +-1.47614 1.41203 +-0.0281192 1.41203 +0.0351436 1.41203 +1.48316 1.41203 +] +[ +-1.47614 1.41906 +-0.0351484 1.41906 +0.0421728 1.41906 +1.48316 1.41906 +] +[ +-1.46911 1.42609 +-0.0351484 1.42609 +0.0421728 1.42609 +1.47613 1.42609 +] +[ +-1.46911 1.43312 +-0.0421776 1.43312 +0.049202 1.43312 +1.47613 1.43312 +] +[ +-1.46208 1.44015 +-0.0421776 1.44015 +0.049202 1.44015 +1.4691 1.44015 +] +[ +-1.45505 1.44718 +-0.0492068 1.44718 +0.0562312 1.44718 +1.46207 1.44718 +] +[ +-1.45505 1.45421 +-0.0492068 1.45421 +0.0562312 1.45421 +1.46207 1.45421 +] +[ +-1.44802 1.46124 +-0.056236 1.46124 +0.0632604 1.46124 +1.45504 1.46124 +] +[ +-1.44099 1.46827 +-0.0632652 1.46827 +0.0702896 1.46827 +1.44801 1.46827 +] +[ +-1.44099 1.4753 +-0.0632652 1.4753 +0.0702896 1.4753 +1.44801 1.4753 +] +[ +-1.43396 1.48233 +-0.0702944 1.48233 +0.0773188 1.48233 +1.44098 1.48233 +] +[ +-1.42693 1.48935 +-0.0773236 1.48935 +0.084348 1.48935 +1.43396 1.48935 +] +[ +-1.42693 1.49638 +-0.0773236 1.49638 +0.084348 1.49638 +1.43396 1.49638 +] +[ +-1.4199 1.50341 +-0.0843528 1.50341 +0.0913772 1.50341 +1.42693 1.50341 +] +[ +-1.41287 1.51044 +-0.091382 1.51044 +0.0984064 1.51044 +1.4199 1.51044 +] +[ +-1.40584 1.51747 +-0.0984112 1.51747 +0.105436 1.51747 +1.41287 1.51747 +] +[ +-1.40584 1.5245 +-0.10544 1.5245 +0.112465 1.5245 +1.41287 1.5245 +] +[ +-1.39881 1.53153 +-0.11247 1.53153 +0.119494 1.53153 +1.40584 1.53153 +] +[ +-1.39178 1.53856 +-0.11247 1.53856 +0.119494 1.53856 +1.39881 1.53856 +] +[ +-1.38476 1.54559 +-0.119499 1.54559 +0.126523 1.54559 +1.39178 1.54559 +] +[ +-1.37773 1.55262 +-0.126528 1.55262 +0.133552 1.55262 +1.38475 1.55262 +] +[ +-1.3707 1.55965 +-0.133557 1.55965 +0.140582 1.55965 +1.37772 1.55965 +] +[ +-1.36367 1.56668 +-0.140586 1.56668 +0.147611 1.56668 +1.37069 1.56668 +] +[ +-1.35664 1.5737 +-0.147616 1.5737 +0.15464 1.5737 +1.36366 1.5737 +] +[ +-1.34961 1.58073 +-0.154645 1.58073 +0.161669 1.58073 +1.35663 1.58073 +] +[ +-1.34258 1.58776 +-0.168703 1.58776 +0.175728 1.58776 +1.3496 1.58776 +] +[ +-1.33555 1.59479 +-0.175732 1.59479 +0.182757 1.59479 +1.34258 1.59479 +] +[ +-1.32852 1.60182 +-0.182762 1.60182 +0.189786 1.60182 +1.33555 1.60182 +] +[ +-1.32149 1.60885 +-0.189791 1.60885 +0.196815 1.60885 +1.32852 1.60885 +] +[ +-1.31446 1.61588 +-0.19682 1.61588 +0.203844 1.61588 +1.32149 1.61588 +] +[ +-1.30743 1.62291 +-0.210878 1.62291 +0.217903 1.62291 +1.31446 1.62291 +] +[ +-1.3004 1.62994 +-0.217908 1.62994 +0.224932 1.62994 +1.30743 1.62994 +] +[ +-1.28635 1.63697 +-0.224937 1.63697 +0.231961 1.63697 +1.29337 1.63697 +] +[ +-1.27932 1.644 +-0.238995 1.644 +0.24602 1.644 +1.28634 1.644 +] +[ +-1.27229 1.65103 +-0.246024 1.65103 +0.253049 1.65103 +1.27931 1.65103 +] +[ +-1.25823 1.65805 +-0.260083 1.65805 +0.267107 1.65805 +1.26525 1.65805 +] +[ +-1.2512 1.66508 +-0.267112 1.66508 +0.274137 1.66508 +1.25822 1.66508 +] +[ +-1.24417 1.67211 +-0.281171 1.67211 +0.288195 1.67211 +1.25119 1.67211 +] +[ +-1.23011 1.67914 +-0.295229 1.67914 +0.302253 1.67914 +1.23714 1.67914 +] +[ +-1.21605 1.68617 +-0.302258 1.68617 +0.309283 1.68617 +1.22308 1.68617 +] +[ +-1.20902 1.6932 +-0.316317 1.6932 +0.323341 1.6932 +1.21605 1.6932 +] +[ +-1.19497 1.70023 +-0.330375 1.70023 +0.337399 1.70023 +1.20199 1.70023 +] +[ +-1.18091 1.70726 +-0.344433 1.70726 +0.351458 1.70726 +1.18793 1.70726 +] +[ +-1.16685 1.71429 +-0.358492 1.71429 +0.365516 1.71429 +1.17387 1.71429 +] +[ +-1.15279 1.72132 +-0.37255 1.72132 +0.379575 1.72132 +1.15981 1.72132 +] +[ +-1.13873 1.72835 +-0.386609 1.72835 +0.393633 1.72835 +1.14576 1.72835 +] +[ +-1.12467 1.73538 +-0.407696 1.73538 +0.414721 1.73538 +1.1317 1.73538 +] +[ +-1.11062 1.7424 +-0.421755 1.7424 +0.428779 1.7424 +1.11764 1.7424 +] +[ +-1.08953 1.74943 +-0.442842 1.74943 +0.449867 1.74943 +1.09655 1.74943 +] +[ +-1.07547 1.75646 +-0.46393 1.75646 +0.470954 1.75646 +1.08249 1.75646 +] +[ +-1.05438 1.76349 +-0.485017 1.76349 +0.492042 1.76349 +1.06141 1.76349 +] +[ +-1.03329 1.77052 +-0.506105 1.77052 +0.513129 1.77052 +1.04032 1.77052 +] +[ +-1.00518 1.77755 +-0.534222 1.77755 +0.541246 1.77755 +1.0122 1.77755 +] +[ +-0.97706 1.78458 +-0.562338 1.78458 +0.569363 1.78458 +0.984084 1.78458 +] +[ +-0.948943 1.79161 +-0.597484 1.79161 +0.604509 1.79161 +0.955967 1.79161 +] +[ +-0.906768 1.79864 +-0.639659 1.79864 +0.646684 1.79864 +0.913792 1.79864 +] +[ +-0.836476 1.80567 +-0.709951 1.80567 +0.716976 1.80567 +0.843501 1.80567 +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] +[ +] diff --git a/graphics/pstricks/contrib/pst-func/doc/pst-func-doc.pdf b/graphics/pstricks/contrib/pst-func/doc/pst-func-doc.pdf Binary files differnew file mode 100644 index 0000000000..e3f65b7c0c --- /dev/null +++ b/graphics/pstricks/contrib/pst-func/doc/pst-func-doc.pdf diff --git a/graphics/pstricks/contrib/pst-func/doc/pst-func-doc.tex b/graphics/pstricks/contrib/pst-func/doc/pst-func-doc.tex new file mode 100644 index 0000000000..bf8628902f --- /dev/null +++ b/graphics/pstricks/contrib/pst-func/doc/pst-func-doc.tex @@ -0,0 +1,2580 @@ +%% $Id: pst-func-doc.tex 861 2018-12-13 20:40:06Z herbert $ +\documentclass[fontsize=11pt,english,BCOR=10mm,DIV=12,bibliography=totoc,parskip=false, + headings=small, headinclude=false,footinclude=false,oneside]{pst-doc} +\usepackage{pst-func} +\let\pstFuncFV\fileversion +\usepackage{pst-math} +\usepackage{pstricks-add} +\usepackage{animate} +\renewcommand\bgImage{% +\psset{yunit=4cm,xunit=3} +\begin{pspicture}(-2,-0.2)(2,1.4) + \psaxes[Dy=0.25]{->}(0,0)(-2,0)(2,1.25)[$x$,0][$y$,90] + \rput[lb](1,0.75){\textcolor{red}{$\sigma =0.5$}} + \rput[lb](1,0.5){\textcolor{blue}{$\sigma =1$}} + \rput[lb](-2,0.5){$f(x)=\dfrac{1}{\sigma\sqrt{2\pi}}\,e^{-\dfrac{(x-\mu)^2}{2\sigma{}^2}}$} + \psGauss[linecolor=red, linewidth=2pt]{-1.75}{1.75}% + \psGaussI[linewidth=1pt]{-2}{2}% + \psGauss[linecolor=cyan, mue=0.5, linewidth=2pt]{-1.75}{1.75}% + \psGauss[sigma=1, linecolor=blue, linewidth=2pt]{-1.75}{1.75} +\end{pspicture}% +} + +%\usepackage[style=dtk]{biblatex} +\addbibresource{pst-func-doc.bib} + + +\lstset{language=PSTricks, + morekeywords={psGammaDist,psChiIIDist,psTDist,psFDist,psBetaDist,psPlotImpl},basicstyle=\footnotesize\ttfamily, + literate=% + {Ö}{{\"O}}1 + {Ä}{{\"A}}1 + {Ü}{{\"U}}1 + {ß}{{\ss}}1 + {ü}{{\"u}}1 + {ä}{{\"a}}1 + {ö}{{\"o}}1 + {~}{{\textasciitilde}}1 +} +% +\psset{labelFontSize=\scriptstyle}% for mathmode +%\def\pshlabel#1{\footnotesize#1} +%\def\psvlabel#1{\footnotesize#1} +% +\begin{document} + +\title{\texttt{pst-func}} +\subtitle{Plotting special mathematical functions; v.\pstFuncFV} +\author{Herbert Vo\ss} +\docauthor{} +\date{\today} +\maketitle + +\tableofcontents +\psset{unit=1cm} + +\clearpage + +\begin{abstract} +\noindent +\LPack{pst-func} loads by default the following packages: \LPack{pst-plot}, +\LPack{pstricks-add}, \LPack{pst-math}, \LPack{pst-xkey}, and, of course \LPack{pstricks}. +All should be already part of your local \TeX\ installation. If not, or in case +of having older versions, go to \url{http://www.CTAN.org/} and load the newest version. + +\vfill\noindent +Thanks to \\ + Rafal Bartczuk, + Jean-C\^ome Charpentier, + Martin Chicoine, + Gerry Coombes, + Denis Girou, + John Frampton, + Leon Free, + Attila Gati, + Horst Gierhardt, + Jürgen Gilg, + Christophe Jorssen, + Lars Kotthoff, + Buddy Ledger, + Manuel Luque, + Patrice Mégret, + Svend Mortensen, + Matthias Rüss, + Thomas Söll, + Jose-Emilio Vila-Forcen, + Timothy Van Zandt, + Michael Zedler, +and last but not least + \url{http://mathworld.wolfram.com}. + +\end{abstract} + + + +\section{\nxLcs{psBezier\#}} +This macro can plot a B\'ezier spline from order $1$ up to $9$ which needs +(order+$1$) pairs of given coordinates. + +Given a set of $n+1$ control points $P_0$, $P_1$, \ldots, $P_n$, +the corresponding \Index{B\'ezier} curve (or \Index{Bernstein-B\'ezier} curve) is given by +% +\begin{align} +C(t)=\sum_{i=0}^n P_i B_{i,n}(t) +\end{align} +% +where $B_{i,n}(t)$ is a Bernstein polynomial $B_{i,n}(t)=\binom{n}{i}t^i(1-t)^{n-i}$, +and $t \in [0,1]$. +The Bézier curve starts through the first and last given point and +lies within the convex hull of all control points. The curve is tangent +to $P_1-P_0$ and $P_n-P_{n-1}$ at the endpoint. +Undesirable properties of \Index{Bézier curve}s are their numerical instability for +large numbers of control points, and the fact that moving a single control +point changes the global shape of the curve. The former is sometimes avoided +by smoothly patching together low-order Bézier curves. + +The macro \Lcs{psBezier} (note the upper case B) expects the number of the order +and $n=\text{order}+1$ pairs of coordinates: + +\begin{BDef} +\Lcs{psBezier}\Larg{\#}\OptArgs\coord0\coord1\coordn +\end{BDef} + + +The number of steps between the first and last control points is given +by the keyword \Lkeyword{plotpoints} and preset to $200$. It can be +changed in the usual way. + +\begin{lstlisting} +\psset{showpoints=true,linewidth=1.5pt} +\begin{pspicture}(-2,-2)(2,2)% order 1 -- linear + \psBezier1{<->}(-2,0)(-2,2) +\end{pspicture}\qquad +% +\begin{pspicture}(-2,-2)(2,2)% order 2 -- quadratric + \psBezier2{<->}(-2,0)(-2,2)(0,2) +\end{pspicture}\qquad +% +\begin{pspicture}(-2,-2)(2,2)% order 3 -- cubic + \psBezier3{<->}(-2,0)(-2,2)(0,2)(2,2) +\end{pspicture}\qquad + +\vspace{1cm} +\begin{pspicture}(-2,-2)(2,2)% order 4 -- quartic + \psBezier4{<->}(-2,0)(-2,2)(0,2)(2,2)(2,0) +\end{pspicture}\qquad +% +\begin{pspicture}(-2,-2)(2,2)% order 5 -- quintic + \psBezier5{<->}(-2,0)(-2,2)(0,2)(2,2)(2,0)(2,-2) +\end{pspicture}\qquad +% +\begin{pspicture}(-2,-2)(2,2)% order 6 + \psBezier6{<->}(-2,0)(-2,2)(0,2)(2,2)(2,0)(2,-2)(0,-2) +\end{pspicture}\qquad + +\vspace{1cm} +\begin{pspicture}(-2,-2)(2,2)% order 7 + \psBezier7{<->}(-2,0)(-2,2)(0,2)(2,2)(2,0)(2,-2)(0,-2)(-2,-2) +\end{pspicture}\qquad +% +\begin{pspicture}(-2,-2)(2,2)% order 8 + \psBezier8{<->}(-2,0)(-2,2)(0,2)(2,2)(2,0)(2,-2)(0,-2)(-2,-2)(-2,0) +\end{pspicture}\qquad +% +\begin{pspicture}(-2,-2)(2,2)% order 9 + \psBezier9{<->}(-2,0)(-2,2)(0,2)(2,2)(2,0)(2,-2)(0,-2)(-2,-2)(-2,0)(0,0) +\end{pspicture} +\end{lstlisting} + +\begingroup +\psset{showpoints=true,linewidth=1.5pt} +\begin{pspicture}(-2,-2)(2,2)% order 1 -- linear + \psBezier1{<->}(-2,0)(-2,2) +\end{pspicture}\qquad +% +\begin{pspicture}(-2,-2)(2,2)% order 2 -- quadratric + \psBezier2{<->}(-2,0)(-2,2)(0,2) +\end{pspicture}\qquad +% +\begin{pspicture}(-2,-2)(2,2)% order 3 -- cubic + \psBezier3{<->}(-2,0)(-2,2)(0,2)(2,2) +\end{pspicture}\qquad + +\vspace{1cm} +\begin{pspicture}(-2,-2)(2,2)% order 4 -- quartic + \psBezier4{<->}(-2,0)(-2,2)(0,2)(2,2)(2,0) +\end{pspicture}\qquad +% +\begin{pspicture}(-2,-2)(2,2)% order 5 -- quintic + \psBezier5{<->}(-2,0)(-2,2)(0,2)(2,2)(2,0)(2,-2) +\end{pspicture}\qquad +% +\begin{pspicture}(-2,-2)(2,2)% order 6 + \psBezier6{<->}(-2,0)(-2,2)(0,2)(2,2)(2,0)(2,-2)(0,-2) +\end{pspicture}\qquad + +\vspace{1cm} +\begin{pspicture}(-2,-2)(2,2)% order 7 + \psBezier7{<->}(-2,0)(-2,2)(0,2)(2,2)(2,0)(2,-2)(0,-2)(-2,-2) +\end{pspicture}\qquad +% +\begin{pspicture}(-2,-2)(2,2)% order 8 + \psBezier8{<->}(-2,0)(-2,2)(0,2)(2,2)(2,0)(2,-2)(0,-2)(-2,-2)(-2,0) +\end{pspicture}\qquad +% +\begin{pspicture}(-2,-2)(2,2)% order 9 + \psBezier9{<->}(-2,0)(-2,2)(0,2)(2,2)(2,0)(2,-2)(0,-2)(-2,-2)(-2,0)(0,0) +\end{pspicture} +\endgroup + +\clearpage +\section{Polynomials} + +\subsection{Chebyshev polynomials} +The polynomials of the first (\Lps{ChebyshevT}) kind are defined through the identity + +\[ T_n(\cos\theta)=\cos(n\theta)\] + +They can be obtained from the generating functions +\begin{align} + g_1(t,x) &= \frac{1-t^2}{1-2xt+t^2}\\ + &= T_0(x)+2\sum_{n=1}^\infty T_n(x)t^n +\end{align} + +and + +\begin{align} + g_2(t,x) &= \frac{1-xt}{1-2xt+t^2}\\ + &= \sum_{n=0}^\infty T_n(x)t^n +\end{align} + +The polynomials of second kind (\Lps{ChebyshevU}) can be generated by + +\begin{align} + g(t,x) &= \frac{1}{1-2xt+t^2}\\ + &= \sum_{n=0}^\infty U_n(x)t^n +\end{align} + +\LPack{pst-func} defines the \TeX-macros \Lcs{ChebyshevT} for the first kind +and \Lcs{ChebyshevU} for the second kind of \Index{Chebyshev polynomials}. +These \TeX-macros cannot be used outside of PostScript, they are only wrappers +for \verb+tx@FuncDict begin ChebyshevT end+ and the same for \Lcs{ChebyshevU}. + +\begin{center} +\bgroup +\psset{arrowscale=1.5,unit=3cm} +\begin{pspicture}(-1.5,-1.5)(1.5,1.5) + \psaxes[ticks=none,labels=none]{->}(0,0)(-1.25,-1.25)(1.25,1.25)% + [Re$\{s_{21}\}$,0][Im$\{s_{21}\}$,90] + \pscircle(0,0){1} + \parametricplot[linecolor=blue,plotpoints=10000]{0}{1.5}{ + /N 9 def + /x 2 N mul t \ChebyshevT def + /y 2 N mul 1 sub t \ChebyshevU def + x x 2 exp y 2 exp add div + y x 2 exp y 2 exp add div + } +\end{pspicture} +\egroup +\end{center} + +\begin{lstlisting} +\psset{arrowscale=1.5,unit=3cm} +\begin{pspicture}(-1.5,-1.5)(1.5,1.5) + \psaxes[ticks=none,labels=none]{->}(0,0)(-1.25,-1.25)(1.25,1.25)% + [Re$\{s_{21}\}$,0][Im$\{s_{21}\}$,90] + \pscircle(0,0){1} + \parametricplot[linecolor=blue,plotpoints=10000]{0}{1.5}{ + /N 9 def + /x 2 N mul t \ChebyshevT def + /y 2 N mul 1 sub t \ChebyshevU def + x x 2 exp y 2 exp add div + y x 2 exp y 2 exp add div + } +\end{pspicture} +\end{lstlisting} + +\begin{center} +\bgroup +\psset{xunit=4cm,yunit=3cm,plotpoints=1000} +\begin{pspicture}(-1.2,-2)(2,1.5) + \psaxes[Dx=0.2]{->}(0,0)(-1.25,-1.2)(1.25,1.2) + \psset{linewidth=1.5pt} + \psplot[linestyle=dashed]{-1}{1}{1 x \ChebyshevT} + \psplot[linecolor=black]{-1}{1}{2 x \ChebyshevT} + \psplot[linecolor=black]{-1}{1}{3 x \ChebyshevT} + \psplot[linecolor=blue]{-1}{1}{4 x \ChebyshevT } + \psplot[linecolor=red]{-1}{1}{5 x \ChebyshevT } +\end{pspicture} +\egroup +\end{center} + +\begin{lstlisting} +\psset{xunit=4cm,yunit=3cm,plotpoints=1000} +\begin{pspicture}(-1.2,-2)(2,1.5) + \psaxes[Dx=0.2]{->}(0,0)(-1.25,-1.2)(1.25,1.2) + \psset{linewidth=1.5pt} + \psplot[linestyle=dashed]{-1}{1}{1 x \ChebyshevT} + \psplot[linecolor=black]{-1}{1}{2 x \ChebyshevT} + \psplot[linecolor=black]{-1}{1}{3 x \ChebyshevT} + \psplot[linecolor=blue]{-1}{1}{4 x \ChebyshevT } + \psplot[linecolor=red]{-1}{1}{5 x \ChebyshevT } +\end{pspicture} +\end{lstlisting} + +\begin{center} +\bgroup +\psset{xunit=4cm,yunit=3cm,plotpoints=1000} +\begin{pspicture*}(-1.5,-1.5)(1.5,1.5) + \psaxes[Dx=0.2]{->}(0,0)(-1.15,-1.1)(1.15,1.1) + \psset{linewidth=1.5pt} + \psplot[linecolor=black]{-1}{1}{2 x \ChebyshevU} + \psplot[linecolor=black]{-1}{1}{3 x \ChebyshevU} + \psplot[linecolor=blue]{-1}{1}{4 x \ChebyshevU } + \psplot[linecolor=red]{-1}{1}{5 x \ChebyshevU } +\end{pspicture*} +\egroup +\end{center} + +\begin{lstlisting} +\psset{xunit=4cm,yunit=3cm,plotpoints=1000} +\begin{pspicture*}(-1.5,-1.5)(1.5,1.5) + \psaxes[Dx=0.2]{->}(0,0)(-1.15,-1.1)(1.15,1.1) + \psaxes[Dx=0.2]{->}(0,0)(-1.25,-1.2)(1.25,1.2) + \psset{linewidth=1.5pt} + \psplot[linecolor=black]{-1}{1}{2 x \ChebyshevU} + \psplot[linecolor=black]{-1}{1}{3 x \ChebyshevU} + \psplot[linecolor=blue]{-1}{1}{4 x \ChebyshevU} + \psplot[linecolor=red]{-1}{1}{5 x \ChebyshevU} +\end{pspicture*} +\end{lstlisting} + +\begin{center} +\bgroup +\psset{xunit=4cm,yunit=3cm,plotpoints=1000} +\begin{pspicture}(-1.25,-1.2)(1.25,1.2) + \psaxes[Dx=0.2]{->}(0,0)(-1.25,-1.1)(1.25,1.1) + \psset{linewidth=1.5pt} + \psplot[linecolor=black]{-1}{1}{x ACOS 2 mul RadtoDeg cos} + \psplot[linecolor=black]{-1}{1}{x ACOS 3 mul RadtoDeg cos} + \psplot[linecolor=blue]{-1}{1}{x ACOS 4 mul RadtoDeg cos} + \psplot[linecolor=red]{-1}{1}{x ACOS 5 mul RadtoDeg cos} +\end{pspicture} +\egroup +\end{center} + +\begin{lstlisting} +\psset{xunit=4cm,yunit=3cm,plotpoints=1000} +\begin{pspicture}(-1.25,-1.2)(1.25,1.2) + \psaxes[Dx=0.2]{->}(0,0)(-1.25,-1.2)(1.25,1.2) + \psset{linewidth=1.5pt} + \psplot[linecolor=black]{-1}{1}{x ACOS 2 mul RadtoDeg cos} + \psplot[linecolor=black]{-1}{1}{x ACOS 3 mul RadtoDeg cos} + \psplot[linecolor=blue]{-1}{1}{x ACOS 4 mul RadtoDeg cos} + \psplot[linecolor=red]{-1}{1}{x ACOS 5 mul RadtoDeg cos} +\end{pspicture} +\end{lstlisting} + +\subsection{\Lcs{psPolynomial}} +The polynomial function is defined as +% +\begin{align} +f(x) &= a_0 + a_1x + a_2x^2 + a_3x^3 + \ldots +a_{n-1}x^{n-1} + a_nx^n\\ +f^{\prime}(x) &= a_1 + 2a_2x + 3a_3x^2 + \ldots +(n-1)a_{n-1}x^{n-2} + na_nx^{n-1}\\ +f^{\prime\prime}(x) &= 2a_2 + 6a_3x + \ldots +(n-1)(n-2)a_{n-1}x^{n-3} + n(n-1)a_nx^{n-2} +\end{align} + +\noindent so \LPack{pst-func} needs only the \Index{coefficients} +of the polynomial to calculate the function. The syntax is + +\begin{BDef} +\Lcs{psPolynomial}\OptArgs\Largb{xStart}\Largb{xEnd} +\end{BDef} + +With the option \Lkeyword{xShift} one can do a horizontal shift to the graph of the function. +With another than the predefined value the macro replaces $x$ by $x-x\mathrm{Shift}$; +\Lkeyword{xShift}=1 moves the graph of the \Index{polynomial function} one unit to the right. + +\begin{center} +\bgroup +\psset{yunit=0.5cm,xunit=1cm} +\begin{pspicture*}(-3,-5)(5,10) + \psaxes[Dy=2]{->}(0,0)(-3,-5)(5,10) + \psset{linewidth=1.5pt} + \psPolynomial[coeff=6 3 -1,linecolor=red]{-3}{5} + \psPolynomial[coeff=2 -1 -1 .5 -.1 .025,linecolor=blue]{-2}{4} + \psPolynomial[coeff=-2 1 -1 .5 .1 .025 .2 ,linecolor=magenta]{-2}{4} + \psPolynomial[coeff=-2 1 -1 .5 .1 .025 .2 ,linecolor=magenta,xShift=1,linestyle=dashed]{-2}{4} + \rput[lb](4,4){\textcolor{red}{$f(x)$}} + \rput[lb](4,8){\textcolor{blue}{$g(x)$}} + \rput[lb](2,4){\textcolor{magenta}{$h(x)$}} +\end{pspicture*} +\egroup +\end{center} + +\begin{lstlisting} +\psset{yunit=0.5cm,xunit=1cm} +\begin{pspicture*}(-3,-5)(5,10) + \psaxes[Dy=2]{->}(0,0)(-3,-5)(5,10) + \psset{linewidth=1.5pt} + \psPolynomial[coeff=6 3 -1,linecolor=red]{-3}{5} + \psPolynomial[coeff=2 -1 -1 .5 -.1 .025,linecolor=blue]{-2}{4} + \psPolynomial[coeff=-2 1 -1 .5 .1 .025 .2 ,linecolor=magenta]{-2}{4} + \psPolynomial[coeff=-2 1 -1 .5 .1 .025 .2 ,linecolor=magenta,xShift=1,linestyle=dashed]{-2}{4} + \rput[lb](4,4){\textcolor{red}{$f(x)$}} + \rput[lb](4,8){\textcolor{blue}{$g(x)$}} + \rput[lb](2,4){\textcolor{magenta}{$h(x)$}} +\end{pspicture*} +\end{lstlisting} + +The plot is easily clipped using the star version of the +\Lenv{pspicture} environment, so that points whose coordinates +are outside of the desired range are not plotted. +The plotted polynomials are: +% +\begin{align} +f(x) & = 6 + 3x -x^2 \\ +g(x) & = 2 -x -x^2 +0.5x^3 -0.1x^4 +0.025x^5\\ +h(x) & = -2 +x -x^2 +0.5x^3 +0.1x^4 +0.025x^5+0.2x^6\\ +h^*(x) & = -2 +(x-1) -(x-1)^2 +0.5(x-1)^3 +\nonumber\\ + & \phantom{ = }+0.1(x-1)^4 +0.025(x-1)^5+0.2(x-1)^6 +\end{align} +% +There are the following new options: + +\noindent\medskip +{\tabcolsep=2pt +\begin{tabularx}{\linewidth}{@{}l>{\ttfamily}l>{\ttfamily}lX@{}} +Name & \textrm{Value} & \textrm{Default}\\\hline +\Lkeyword{coeff} & a0 a1 a2 ... & 0 0 1 & The coefficients must have the order $a_0\ a_1\ a_2 \ldots$ and +be separated by \textbf{spaces}. The number of coefficients +is limited only by the memory of the computer ... The default +value of the parameter \Lkeyword{coeff} is \verb+0 0 1+, which gives +the parabola $y=a_0+a_1x+a_2x^2=x^2$.\\ +\Lkeyword{xShift} & <number> & 0 & $(x-xShift)$ for the horizontal shift of the polynomial\\ +\Lkeyword{Derivation} & <number> & 0 & the default is the function itself\\ +\Lkeyword{markZeros} & false|true & false & dotstyle can be changed\\ +\Lkeyword{epsZero} & <value> & 0.1 & The distance between two zeros, important for + the iteration function to test, if the zero value still + exists\\ +\Lkeyword{dZero} & <value> & 0.1 & When searching for all zero values, the function is scanned + with this step\\ +\Lkeyword{zeroLineTo} & <number> & false & plots a line from the zero point to the value of the + zeroLineTo's Derivation of the polynomial function\\ +\Lkeyword{zeroLineStyle} & <line style> & \Lkeyval{dashed} & the style is one of the for \PST valid styles.\\ +\Lkeyword{zeroLineColor} & <color> & \Lkeyval{black} & any valid xolor is possible\\ +\Lkeyword{zeroLineWidth} & <width> & \rlap{0.5\textbackslash pslinewidth} & \\ +\end{tabularx} +} + +\bigskip +The above parameters are only +valid for the \Lcs{psPolynomial} macro, except \verb+x0+, which can also be used for the Gauss function. All +options can be set in the usual way with \Lcs{psset}. + +\bigskip +\begin{LTXexample} +\psset{yunit=0.5cm,xunit=2cm} +\begin{pspicture*}(-3,-5)(3,10) + \psaxes[Dy=2]{->}(0,0)(-3,-5)(3,10) + \psset{linewidth=1.5pt} + \psPolynomial[coeff=-2 1 -1 .5 .1 .025 .2 ,linecolor=magenta]{-2}{4} + \psPolynomial[coeff=-2 1 -1 .5 .1 .025 .2 ,linecolor=red,% + linestyle=dashed,Derivation=1]{-2}{4} + \psPolynomial[coeff=-2 1 -1 .5 .1 .025 .2 ,linecolor=blue,% + linestyle=dotted,Derivation=2]{-2}{4} + \rput[lb](2,4){\textcolor{magenta}{$h(x)$}} + \rput[lb](1,1){\textcolor{red}{$h^{\prime}(x)$}} + \rput[lb](-1,6){\textcolor{blue}{$h^{\prime\prime}(x)$}} +\end{pspicture*} +\end{LTXexample} +%$ +\begin{LTXexample} +\psset{yunit=0.5cm,xunit=2cm} +\begin{pspicture*}(-3,-5)(3,10) + \psaxes[Dy=2]{->}(0,0)(-3,-5)(3,10) + \psset{linewidth=1.5pt} + \psPolynomial[coeff=0 0 0 1,linecolor=blue]{-2}{4} + \psPolynomial[coeff=0 0 0 1,linecolor=red,% + linestyle=dashed,Derivation=2]{-2}{4} + \psPolynomial[coeff=0 0 0 1,linecolor=cyan,% + linestyle=dotted,Derivation=3]{-2}{4} + \rput[lb](1.8,4){\textcolor{blue}{$f(x)=x^3$}} + \rput[lb](0.2,8){\textcolor{red}{$f^{\prime\prime}(x)=6x$}} + \rput[lb](-2,5){\textcolor{cyan}{$f^{\prime\prime\prime}(x)=6$}} +\end{pspicture*} +\end{LTXexample} +%$ +\begin{LTXexample} +\begin{pspicture*}(-5,-5)(5,5) + \psaxes{->}(0,0)(-5,-5)(5,5)% + \psset{dotscale=2} + \psPolynomial[markZeros,linecolor=red,linewidth=2pt,coeff=-1 1 -1 0 0.15]{-4}{3}% + \psPolynomial[markZeros,linecolor=blue,linewidth=1pt,linestyle=dashed,% + coeff=-1 1 -1 0 0.15,Derivation=1,zeroLineTo=0]{-4}{3}% + \psPolynomial[markZeros,linecolor=magenta,linewidth=1pt,linestyle=dotted,% + coeff=-1 1 -1 0 0.15,Derivation=2,zeroLineTo=0]{-4}{3}% + \psPolynomial[markZeros,linecolor=magenta,linewidth=1pt,linestyle=dotted,% + coeff=-1 1 -1 0 0.15,Derivation=2,zeroLineTo=1]{-4}{3}% +\end{pspicture*} +\end{LTXexample} + +\begin{LTXexample} +\psset{xunit=1.5} +\begin{pspicture*}(-5,-5)(5,5) + \psaxes{->}(0,0)(-5,-5)(5,5)% + \psset{dotscale=2,dotstyle=x,zeroLineStyle=dotted,zeroLineWidth=1pt} + \psPolynomial[markZeros,linecolor=red,linewidth=2pt,coeff=-1 1 -1 0 0.15]{-4}{3}% + \psPolynomial[markZeros,linecolor=blue,linewidth=1pt,linestyle=dashed,% + coeff=-1 1 -1 0 0.15,Derivation=1,zeroLineTo=0]{-4}{3}% + \psPolynomial[markZeros,linecolor=magenta,linewidth=1pt,linestyle=dotted,% + coeff=-1 1 -1 0 0.15,Derivation=2,zeroLineTo=0]{-4}{3}% + \psPolynomial[markZeros,linecolor=magenta,linewidth=1pt,linestyle=dotted,% + coeff=-1 1 -1 0 0.15,Derivation=2,zeroLineTo=1]{-4}{3}% +\end{pspicture*} +\end{LTXexample} + + +\clearpage +\subsection{\Lcs{psBernstein}} +The polynomials defined by +% +\[ B_{i,n}(t)=\binom{n}{i}t^i(1-t)^{n-i} \] +% +where $\tbinom{n}{k}$ is a binomial coefficient are named Bernstein polynomials of degree $n$. +They form a basis for the power polynomials of degree $n$. +The Bernstein polynomials satisfy symmetry +\[B_{i,n}(t)=B_{n-i,n}(1-t)\] +positivity \[B_{i,n}(t)\ge0 \mbox{\qquad for } 0\le t\le1\] +normalization \[\sum_{i=0}^nB_{i,n}(t)=1\] +and $B_{i,n}$ with $i!=0$, $n$ has a single unique local maximum of +\[i^in^{-n}(n-i)^{n-i}\binom{n}{i}\] +occurring at $t=\frac{i}{n}$. +The envelope $f_n(x)$ of the Bernstein polynomials $B_{i,n}(x)$ for $i=0,1,\ldots,n$ +is given by \[f_n(x)=\frac{1}{\sqrt{\pi n\cdot x(1-x)}}\] +illustrated below for $n=20$. + +\begin{BDef} +\Lcs{psBernstein}\OptArgs\Largr{tStart,tEnd}\Largr{i,n} +\end{BDef} + +The (\Lkeyword{tStart}, \Lkeyword{tEnd}) are \emph{optional} and preset by \verb=(0,1)=. +The only new optional argument is the boolean key \Lkeyword{envelope}, +which plots the envelope curve instead of the Bernstein polynomial. + +\begin{LTXexample}[width=5cm,pos=l] +\psset{xunit=4.5cm,yunit=3cm} +\begin{pspicture}(1,1.1) + \psaxes{->}(0,0)(1,1)[$t$,0][$B_{0,0}$,90] + \psBernstein[linecolor=red,linewidth=1pt](0,0) +\end{pspicture} +\end{LTXexample} + +\begin{LTXexample}[width=5cm,pos=l] +\psset{xunit=4.5cm,yunit=3cm} +\begin{pspicture}(1,1.1) + \psaxes{->}(0,0)(1,1)[$t$,0][$B_{i,1}$,90] + \psBernstein[linecolor=blue,linewidth=1pt](0,1) + \psBernstein[linecolor=blue,linewidth=1pt](1,1) +\end{pspicture} +\end{LTXexample} + +\begin{LTXexample}[width=5cm,pos=l] +\psset{xunit=4.5cm,yunit=3cm} +\begin{pspicture}(1,1.1) + \psaxes{->}(0,0)(1,1)[$t$,0][$B_{i,2}$,90] + \multido{\i=0+1}{3}{\psBernstein[linecolor=red, + linewidth=1pt](\i,2)} +\end{pspicture} +\end{LTXexample} + +\begin{LTXexample}[width=5cm,pos=l] +\psset{xunit=4.5cm,yunit=3cm} +\begin{pspicture}(1,1.1) + \psaxes{->}(0,0)(1,1)[$t$,0][$B_{i,3}$,90] + \multido{\i=0+1}{4}{\psBernstein[linecolor=magenta, + linewidth=1pt](\i,3)} +\end{pspicture} +\end{LTXexample} + +\begin{LTXexample}[width=5cm,pos=l] +\psset{xunit=4.5cm,yunit=3cm} +\begin{pspicture}(1,1.1) + \psaxes{->}(0,0)(1,1)[$t$,0][$B_{i,4}$,90] + \multido{\i=0+1}{5}{\psBernstein[linecolor=cyan, + linewidth=1pt](\i,4)} +\end{pspicture} +\end{LTXexample} + +\begin{LTXexample}[width=5cm,pos=l] +\psset{xunit=4.5cm,yunit=3cm} +\begin{pspicture}(-0.1,-0.05)(1.1,1.1) + \multido{\i=0+1}{20}{\psBernstein[linecolor=green, + linewidth=1pt](\i,20)} + \psBernstein[envelope,linecolor=black](0.02,0.98)(0,20) + \psaxes{->}(0,0)(1,1)[$t$,0][$B_{i,20}$,180] +\end{pspicture} +\end{LTXexample} + +\begin{LTXexample}[width=5cm,pos=l] +\psset{xunit=4.5cm,yunit=3cm} +\begin{pspicture*}(-0.2,-0.05)(1.1,1.1) + \psaxes{->}(0,0)(1,1)[$t$,0][$B_{env}$,180] + \multido{\i=2+1}{20}{\psBernstein[envelope, + linewidth=1pt](0.01,0.99)(0,\i)} +\end{pspicture*} +\end{LTXexample} + + +\clearpage +\section{Calculating the zeros of a function or the the intermediate point of two function} + +\begin{BDef} +\Lcs{psZero}\OptArgs\Largr{$x_0,x_1$}\Largb{functionA}\OptArg{functionB}\Largb{node name} +\end{BDef} + +If the second function is not given the macro calculates and displays the zeros of +the first function. If the second function is defined too, then the macro calculates the +intermediate point of the two functions. The intervall is defined as $[x_0,x_1]$. +Possible optional arguments are + + +\medskip +\begin{tabularx}{\linewidth}{ @{} l >{\ttfamily}l X @{} }\toprule +\emph{Name} & \emph{Default} & \emph{Meaning} \\\midrule +\Lkeyword{markZeros} & false & Mark the zeros/intermediate points with a symbol.\\ +\Lkeyword{Newton} & false & Use Newton method instead of the bisector one.\\ +\Lkeyword{PrintCoord} & false & Print the pair of coordinates of the zero/intermediate point, like $P(x|y)$.\\ +\Lkeyword{onlyNode} & false & Calculate only the node, do not print anything, if markZeros $=$ false.\\ +\Lkeyword{onlyYVal} & false & Print only the $y$-value.\\ +\Lkeyword{xory} & false & Print $x=$ $x$-Value or, if onlyYVal $=$ true, $y=$ $y$-value.\\ +\Lkeyword{approx} & true & Change the $=$, if xory $=$ true to $\approx$.\\ +\Lkeyword{originV} & false & Put the values without an offset.\\ +\Lkeyword{Framed} & false & Show a filled frame in backround, framesep, fillcolor, opacity or + linestyle are options to show different frames.\\ +\Lkeyword{PointName} & I & The printed prefix for the calculated Points.\\ +\Lkeyword{decimals} & 2 & The decimals for the $x$ value.\\ +\Lkeyword{ydecimals} & 2 & The decimals for the $y$ value.\\ +\Lkeyword{xShift} & 0 & $x$ move for the printed value.\\ +\Lkeyword{yShift} & 0 & $y$ move for the printed value.\\ +\bottomrule +\end{tabularx} + +\medskip +The following examples where done by Jürgen Gilg and Thomas Söll. + +\bigskip +\definecolor{BeigeTS}{rgb}{0.98,0.95,0.87} +\definecolor{CornBlauTS}{rgb}{0.39,0.59,0.93} +\definecolor{SandBraun}{rgb}{0.96,0.64,0.38} +\psset{yunit=1.25cm,arrowinset=0.02,arrowlength=2,linewidth=0.5pt,saveNodeCoors,NodeCoorPrefix=n,comma} +\def\funkf{2*sqrt(x)*cos(ln(x))*sin(x)} +\begin{pspicture}[plotpoints=500,algebraic,fontscale=5,markZeros, + PointName=N,dotscale=0.7](-0.5,-3)(10,2.5) +\psStep[fillstyle=solid,fillcolor=BeigeTS,opacity=0.7,linewidth=0.3pt, + linecolor=SandBraun!50](0.001,9.5){40}{\funkf} +\psStep[StepType=Riemann,fillstyle=solid,opacity=0.3,fillcolor=CornBlauTS, + linecolor=CornBlauTS,linewidth=0.3pt](0.001,9.5){40}{\funkf} +\psaxes[labelFontSize=\scriptstyle,ticksize=-0.1 0]{->}(0,0)(0,-2.75)(10,2.5) +\psplot[linecolor=BeigeTS!60,linewidth=0.8pt]{0.001}{9.75}{\funkf} +\psplotTangent[linecolor=blue,Derive={Derive(1,\funkf)}]{1.29}{1.5}{\funkf} +\uput[90](6,1.2){$f(x)=2\cdot\sqrt{x}\cdot\cos{(\ln{x})}\cdot\sin{x}$} +{\psset{dotscale=1.5,linecolor=blue!50!black!90,ydecimals=0,Framed,opacity=0.8,decimals=1,PrintCoord} + \psZero[xShift=-0.2,yShift=0.15,postString=1,Newton](0.5,1){\funkf}{N1} + \psZero[xShift=-0.05,yShift=0.15,postString=2](2,4){\funkf}{N2} + \psZero[xShift=-0.45,yShift=0.15,postString=3](4,6){\funkf}{N3} + \psZero[xShift=-0.45,yShift=0.15,postString=4](6,7){\funkf}{N4} + \psZero[xShift=-0.25,yShift=0.15,PointName=x,postString=5,xory,PrintCoord=false,linestyle=none,fillcolor=green,opacity=0.6](9,11){\funkf}{N5} + \psZero[xShift=-0.95,yShift=0,PointName=M,decimals=0,linestyle=none,fillcolor=SandBraun, + ydecimals=1,opacity=0.8,postString={m=1}](0.5,2){Derive(1,\funkf)-1+\funkf}[\funkf]{M}% +} +\pcline{->}(0.5,-1)(M) +\nbput[nrot=:U,labelsep=0.3,npos=0.2]{% + \scriptsize \psZero[originV=true,xory=true,onlyYVal=true,PointName=f(x),postString={m=1},Framed, + opacity=0.8,linestyle=none,markZeros=false,fontscale=10](0.5,2){Derive(1,\funkf)-1+\funkf}[\funkf]{R}} +\psdot[linecolor=green,strokeopacity=0.8](M) +\uput{0.5}[40](M){\psZero[originV=true,approx=false,xory=true,onlyYVal=true, + PointName=m,postString={m=1},markZeros=false,fontscale=8](0.5,2){Derive(1,\funkf)-1}[1]{R}} +\end{pspicture} + + +%\begin{LTXexample}[pos=t] +\begin{lstlisting} +\definecolor{BeigeTS}{rgb}{0.98,0.95,0.87} +\definecolor{CornBlauTS}{rgb}{0.39,0.59,0.93} +\definecolor{SandBraun}{rgb}{0.96,0.64,0.38} +\psset{yunit=1.25cm,arrowinset=0.02,arrowlength=2,linewidth=0.5pt,saveNodeCoors,NodeCoorPrefix=n,comma} +\def\funkf{2*sqrt(x)*cos(ln(x))*sin(x)} +\begin{pspicture}[plotpoints=500,algebraic,fontscale=5,markZeros,PrintCoord, + PointName=N,dotscale=0.7](-0.5,-3)(10,2.5) +\psStep[fillstyle=solid,fillcolor=BeigeTS,opacity=0.7,linewidth=0.3pt, + linecolor=SandBraun!50](0.001,9.5){40}{\funkf} +\psStep[StepType=Riemann,fillstyle=solid,opacity=0.3,fillcolor=CornBlauTS, + linecolor=CornBlauTS,linewidth=0.3pt](0.001,9.5){40}{\funkf} +\psaxes[labelFontSize=\scriptstyle,ticksize=-0.1 0]{->}(0,0)(0,-2.75)(10,2.5) +\psplot[linecolor=BeigeTS!60,linewidth=0.8pt]{0.001}{9.75}{\funkf} +\psplotTangent[linecolor=blue,Derive={Derive(1,\funkf)}]{1.29}{1.5}{\funkf} +\uput[90](6,1.2){$f(x)=2\cdot\sqrt{x}\cdot\cos{(\ln{x})}\cdot\sin{x}$} +{\psset{dotscale=1.5,linecolor=blue!50!black!90,ydecimals=0,Framed,opacity=0.8,decimals=1} + \psZero[xShift=-0.2,yShift=0.15,postString=1,Newton](0.5,1){\funkf}{N1} + \psZero[xShift=-0.05,yShift=0.15,postString=2](2,4){\funkf}{N2} + \psZero[xShift=-0.45,yShift=0.15,postString=3](4,6){\funkf}{N3} + \psZero[xShift=-0.45,yShift=0.15,postString=4](6,7){\funkf}{N4} + \psZero[xShift=-0.45,yShift=0.15,postString=5](9,11){\funkf}{N5} + \psZero[xShift=-1.15,yShift=0,PointName=M,decimals=0,linestyle=none,fillcolor=SandBraun, + opacity=0.8,postString={m=1}](0.5,2){Derive(1,\funkf)-1+\funkf}[\funkf]{M}% +} +\pcline{->}(0.5,-1)(M) +\nbput[nrot=:U,labelsep=0.01]{% + \scriptsize Steigung ist hier\phantom{i} + \psPrintValueNew[PSfont=Palatino-Roman,decimals=0,round,fontscale=7]{nMx,{Derive(1,\funkf)}}} +\psdot[linecolor=green,strokeopacity=0.8](*{nMx} {\funkf}) +\uput[90](*{nMx} {\funkf}){$m=$ + \psPrintValueNew[PSfont=Palatino-Roman,decimals=0,round,fontscale=8]{nMx,{Derive(1,\funkf)}}} +\end{pspicture} +\end{lstlisting} +%\end{LTXexample} + +{\psset{yunit=0.8,comma,decimals=2,algebraic=true,markZeros=true,plotpoints=500,saveNodeCoors,NodeCoorPrefix=n} +%----------------- FUNKTIONSDEFINITIONEN in "algebraic" ----------------- +\def\funkf{0.75*x^4-3*x^2-2} +\def\funkg{0.25*x+1} + +\begin{pspicture}(-6.5,-5.5)(6.5,8.5) +%------ Gitter im Hintergrund (CLIPPED) ----------------- +\begin{psclip}% +{\psframe[linestyle=none](-6.4,-5.4)(6.4,7.4)} +\psgrid[subgriddiv=2,gridlabels=0,gridwidth=0.3pt,gridcolor=black!50,subgridwidth=0.2pt,subgridcolor=black!30](-6.5,-7.5)(6.5,8.5) +\end{psclip} +%--------- Achsen ------------ +\psaxes[xDecimals=0, yDecimals=0,labelFontSize=\scriptstyle,arrowscale=1.3,arrowinset=0.05,arrowlength=1.9, Dy=1,dy=1,dx=1,Dx=1,subticks=0,comma,tickwidth=0.5pt]{->}(0,0)(-6.5,-5.5)(6.5,7.5)[$x$,-90][$y$,180]% Achsen +%----- Funktionsgraphen plotten (Clippen, damit sie nicht aus dem Gitter ragen) ----------------- +\begin{psclip}% +{\psframe[linestyle=none](-6.5,-5.4)(6.5,7.4)} +\psplot[linewidth=1pt,linecolor=Gray]{-6.5}{6.5}{\funkf}% +\psplot[linewidth=1pt,linecolor=BrickRed]{-6.5}{6.5}{\funkg}% +\end{psclip} +%----------------- SPEZIELLE PUNKTE ----------------- +{\psset{fontscale=8,PrintCoord=true,linestyle=none,opacity=0.8,Framed=true,fillcolor=cyan!10} +%----------------- NULLSTELLEN ----------------- +\psZero[xShift=-0.9,yShift=0.15,PointName={N},postString={1},ydecimals=0](-3,-2){\funkf}[0]{N1} +\psZero[xShift=-0.9,yShift=0.15,PointName={N},postString={2},ydecimals=0](2,3){\funkf}[0]{N2} +%----------------- EXTREMWERTE ----------------- +\psZero[xShift=-0.9,yShift=-0.25,PointName={T},postString={1}](-2,0){Derive(1,\funkf)+\funkf}[\funkf]{T1} +\psZero[xShift=-0.9,yShift=0.25,PointName={H},postString={}](-1,1){Derive(1,\funkf)+\funkf}[\funkf]{H} +\psZero[xShift=-0.9,yShift=-0.25,PointName={T},postString={2}](0,2.5){Derive(1,\funkf)+\funkf}[\funkf]{T2} +%----------------- WENDEPUNKTE ----------------- +\psZero[xShift=-1.2,yShift=-0.25,PointName={W},postString={1}](-1.5,-0.5){Derive(2,\funkf)+\funkf}[\funkf]{W1} +\psZero[xShift=-0.6,yShift=-0.25,PointName={W},postString={2}](0.5,1.5){Derive(2,\funkf)+\funkf}[\funkf]{W2} +\psZero[onlyNode=true,markZeros=false](-1.5,-0.5){Derive(2,\funkf)+Derive(1,\funkf)}[Derive(1,\funkf)]{mW1}%Steigung Wendepunkt 1 ist "nmW1y" +} +%----------------- GLEICHUNG WENDETANGENTE ----------------- +\def\funkWende{nmW1y*(x-nW1x)+nW1y} +%----------------- GLEICHUNG WENDENORMALE ----------------- +\def\funkNormal{-1/nmW1y*(x-nW1x)+nW1y} %m_n=-1/m_t +%----------------- Tangente und Normale in W1 plotten ------------------ +\psplot[linewidth=1pt,linecolor=blue]{-1.3}{2.55}{\funkWende}% +\psplot[linewidth=1pt,linecolor=Green]{-6.5}{5}{\funkNormal}% +%----------------- Punkte und Werte NICHT anzeigen +{\psset{onlyNode=true,markZeros=false} +%----------------- Schnittpunkt: Wendetangente in W1 mit f ------------- +\psZero(0,4){\funkWende}[\funkf]{WS1} +%----------------- Schnittpunkte: Wendenormale in W1 mit f ------------- +\psZero(-4,0){\funkNormal}[\funkf]{WN1} +\psZero(0,1.5){\funkNormal}[\funkf]{WN2} +\psZero(1.5,3){\funkNormal}[\funkf]{WN3} +%----------------- NULLSTELLE von g ----------------- +\psZero(-3,3){\funkg}[0]{Ng1} +%----------------- SCHNITTPUNKTE f und g ----------------- +\psZero(0,3){\funkg}[\funkf]{S1} +\psZero(-3,0){\funkg}[\funkf]{S2} +} +%----------------- FLÄCHE mit x-ACHSE ----------------- +\pscustom[fillstyle=solid,opacity=0.3,fillcolor=gray,linestyle=none]{% +\psplot{nN1x}{nW1x}{\funkf} +\lineto(!nW1x 0) +\closepath +} +%----------------- FLÄCHE ZWISCHEN WENDETANGENTE UND KURVE f ----------------- +\pscustom[fillstyle=solid,opacity=0.3,fillcolor=blue,linestyle=none]{% +\psplot{nW1x}{nWS1x}{\funkWende} +\psplot{nWS1x}{nW1x}{\funkf} +\closepath +} +%----------------- FLÄCHE ZWISCHEN WENDENORMALE UND KURVE f (Zwei FlÄchenstücke!!!) ---- +%----------------- linke FLÄCHE ----------------- +\pscustom[fillstyle=solid,opacity=0.3,fillcolor=green,linestyle=none]{% +\psplot{nWN1x}{nW1x}{\funkNormal} +\psplot{nW1x}{nWN1x}{\funkf} +\closepath +} +%----------------- rechte FLÄCHE ----------------- +\pscustom[fillstyle=solid,opacity=0.3,fillcolor=green,linestyle=none]{% +\psplot{nWN2x}{nWN3x}{\funkNormal} +\psplot{nWN3x}{nWN2x}{\funkf} +\closepath +} +%----------------- FLÄCHE zwischen den KURVEN f und g und beiden KOORDINATEN-ACHSEN ----- +\pscustom[fillstyle=solid,opacity=0.3,fillcolor=yellow,linestyle=none]{% +\psplot{0}{nS1x}{\funkg} +\psplot{nS1x}{nN2x}{\funkf} +\lineto(0,0) +\closepath +} +% SPIELEREI: FLÄCHE mit f und PARALLELEN ZUR x-ACHSE +% Punkte und Werte NICHT anzeigen +{\psset{onlyNode=true,markZeros=false} +\psZero(-3,-2){\funkf}[2]{M1} +\psZero(-3,-2){\funkf}[4]{M2} +} +\pscustom[fillstyle=solid,opacity=0.3,fillcolor=magenta,linestyle=none]{% +\psplot{nM1x}{nM2x}{\funkf} +\lineto(0,4) +\lineto(0,2) +\closepath +} +\end{pspicture}} + +\begin{lstlisting} +\psset{yunit=0.8,comma,decimals=2,algebraic=true,markZeros=true,plotpoints=500,saveNodeCoors,NodeCoorPrefix=n} +%----------------- FUNKTIONSDEFINITIONEN in "algebraic" ----------------- +\def\funkf{0.75*x^4-3*x^2-2} +\def\funkg{0.25*x+1} + +\begin{pspicture}(-6.5,-5.5)(6.5,8.5) +%------ Gitter im Hintergrund (CLIPPED) ----------------- +\begin{psclip}% +{\psframe[linestyle=none](-6.4,-5.4)(6.4,7.4)} +\psgrid[subgriddiv=2,gridlabels=0,gridwidth=0.3pt,gridcolor=black!50,subgridwidth=0.2pt,subgridcolor=black!30](-6.5,-7.5)(6.5,8.5) +\end{psclip} +%--------- Achsen ------------ +\psaxes[xDecimals=0, yDecimals=0,labelFontSize=\scriptstyle,arrowscale=1.3,arrowinset=0.05,arrowlength=1.9, Dy=1,dy=1,dx=1,Dx=1,subticks=0,comma,tickwidth=0.5pt]{->}(0,0)(-6.5,-5.5)(6.5,7.5)[$x$,-90][$y$,180]% Achsen +%----- Funktionsgraphen plotten (Clippen, damit sie nicht aus dem Gitter ragen) ----------------- +\begin{psclip}% +{\psframe[linestyle=none](-6.5,-5.4)(6.5,7.4)} +\psplot[linewidth=1pt,linecolor=Gray]{-6.5}{6.5}{\funkf}% +\psplot[linewidth=1pt,linecolor=BrickRed]{-6.5}{6.5}{\funkg}% +\end{psclip} +%----------------- SPEZIELLE PUNKTE ----------------- +{\psset{fontscale=8,PrintCoord=true,linestyle=none,opacity=0.8,Framed=true,fillcolor=cyan!10} +%----------------- NULLSTELLEN ----------------- +\psZero[xShift=-0.9,yShift=0.15,PointName={N},postString={1},ydecimals=0](-3,-2){\funkf}[0]{N1} +\psZero[xShift=-0.9,yShift=0.15,PointName={N},postString={2},ydecimals=0](2,3){\funkf}[0]{N2} +%----------------- EXTREMWERTE ----------------- +\psZero[xShift=-0.9,yShift=-0.25,PointName={T},postString={1}](-2,0){Derive(1,\funkf)+\funkf}[\funkf]{T1} +\psZero[xShift=-0.9,yShift=0.25,PointName={H},postString={}](-1,1){Derive(1,\funkf)+\funkf}[\funkf]{H} +\psZero[xShift=-0.9,yShift=-0.25,PointName={T},postString={2}](0,2.5){Derive(1,\funkf)+\funkf}[\funkf]{T2} +%----------------- WENDEPUNKTE ----------------- +\psZero[xShift=-1.2,yShift=-0.25,PointName={W},postString={1}](-1.5,-0.5){Derive(2,\funkf)+\funkf}[\funkf]{W1} +\psZero[xShift=-0.6,yShift=-0.25,PointName={W},postString={2}](0.5,1.5){Derive(2,\funkf)+\funkf}[\funkf]{W2} +\psZero[onlyNode=true,markZeros=false](-1.5,-0.5){Derive(2,\funkf)+Derive(1,\funkf)}[Derive(1,\funkf)]{mW1}%Steigung Wendepunkt 1 ist "nmW1y" +} +%----------------- GLEICHUNG WENDETANGENTE ----------------- +\def\funkWende{nmW1y*(x-nW1x)+nW1y} +%----------------- GLEICHUNG WENDETANGENTE ----------------- +\def\funkNormal{-1/nmW1y*(x-nW1x)+nW1y} %m_n=-1/m_t +%----------------- Tangente und Normale in W1 plotten ------------------ +\psplot[linewidth=1pt,linecolor=blue]{-1.3}{2.55}{\funkWende}% +\psplot[linewidth=1pt,linecolor=Green]{-6.5}{5}{\funkNormal}% +%----------------- Punkte und Werte NICHT anzeigen +{\psset{onlyNode=true,markZeros=false} +%----------------- Schnittpunkt: Wendetangente in W1 mit f ------------- +\psZero(0,4){\funkWende}[\funkf]{WS1} +%----------------- Schnittpunkte: Wendenormale in W1 mit f ------------- +\psZero(-4,0){\funkNormal}[\funkf]{WN1} +\psZero(0,1.5){\funkNormal}[\funkf]{WN2} +\psZero(1.5,3){\funkNormal}[\funkf]{WN3} +%----------------- NULLSTELLE von g ----------------- +\psZero(-3,3){\funkg}[0]{Ng1} +%----------------- SCHNITTPUNKTE f und g ----------------- +\psZero(0,3){\funkg}[\funkf]{S1} +\psZero(-3,0){\funkg}[\funkf]{S2} +} +%----------------- FLÄCHE mit x-ACHSE ----------------- +\pscustom[fillstyle=solid,opacity=0.3,fillcolor=gray,linestyle=none]{% +\psplot{nN1x}{nW1x}{\funkf} +\lineto(!nW1x 0) +\closepath +} +%----------------- FLÄCHE ZWISCHEN WENDETANGENTE UND KURVE f ----------------- +\pscustom[fillstyle=solid,opacity=0.3,fillcolor=blue,linestyle=none]{% +\psplot{nW1x}{nWS1x}{\funkWende} +\psplot{nWS1x}{nW1x}{\funkf} +\closepath +} +%----------------- FLÄCHE ZWISCHEN WENDENORMALE UND KURVE f (Zwei FlÄchenstücke!!!) ---- +%----------------- linke FLÄCHE ----------------- +\pscustom[fillstyle=solid,opacity=0.3,fillcolor=green,linestyle=none]{% +\psplot{nWN1x}{nW1x}{\funkNormal} +\psplot{nW1x}{nWN1x}{\funkf} +\closepath +} +%----------------- rechte FLÄCHE ----------------- +\pscustom[fillstyle=solid,opacity=0.3,fillcolor=green,linestyle=none]{% +\psplot{nWN2x}{nWN3x}{\funkNormal} +\psplot{nWN3x}{nWN2x}{\funkf} +\closepath +} +%----------------- FLÄCHE zwischen den KURVEN f und g und beiden KOORDINATEN-ACHSEN ----- +\pscustom[fillstyle=solid,opacity=0.3,fillcolor=yellow,linestyle=none]{% + \psplot{0}{nS1x}{\funkg} + \psplot{nS1x}{nN2x}{\funkf} + \lineto(0,0) + \closepath} +% SPIELEREI: FLÄCHE mit f und PARALLELEN ZUR x-ACHSE +% Punkte und Werte NICHT anzeigen +{\psset{onlyNode=true,markZeros=false} +\psZero(-3,-2){\funkf}[2]{M1} +\psZero(-3,-2){\funkf}[4]{M2}} +\pscustom[fillstyle=solid,opacity=0.3,fillcolor=magenta,linestyle=none]{% + \psplot{nM1x}{nM2x}{\funkf} + \lineto(0,4) + \lineto(0,2) + \closepath} +\end{pspicture} +\end{lstlisting} + + + +%\begin{LTXexample}[pos=t] +\begin{lstlisting} +\definecolor{BeigeTS}{rgb}{0.98,0.95,0.87} +\definecolor{CornBlauTS}{rgb}{0.39,0.59,0.93} +\definecolor{SandBraun}{rgb}{0.96,0.64,0.38} +\psset{yunit=1.25cm,arrowinset=0.02,arrowlength=2,linewidth=0.5pt,saveNodeCoors,NodeCoorPrefix=n} +\def\funkf{2*sqrt(x)*cos(ln(x))*sin(x)} +\begin{pspicture}[plotpoints=500,algebraic,fontscale=5,markZeros,PrintCoord, + PointName=N,dotscale=0.7](-0.5,-3)(10,2.5) +\psStep[fillstyle=solid,fillcolor=BeigeTS,opacity=0.7,linewidth=0.3pt, + linecolor=SandBraun!50](0.001,9.5){40}{\funkf} +\psStep[StepType=Riemann,fillstyle=solid,opacity=0.3,fillcolor=CornBlauTS, + linecolor=CornBlauTS,linewidth=0.3pt](0.001,9.5){40}{\funkf} +\psaxes[labelFontSize=\scriptstyle,ticksize=-0.1 0]{->}(0,0)(0,-2.75)(10,2.5) +\psplot[linecolor=BeigeTS!60,linewidth=0.8pt]{0.001}{9.75}{\funkf} +\psplotTangent[linecolor=blue,Derive={Derive(1,\funkf)}]{1.29}{1.5}{\funkf} +\uput[90](6,1.2){$f(x)=2\cdot\sqrt{x}\cdot\cos{(\ln{x})}\cdot\sin{x}$} +{\psset{dotscale=1.5,linecolor=blue!50!black!90,ydecimals=0} + \psZero[xShift=-0.2,yShift=0.15,postString=1,Newton](0.5,1){\funkf}{N1} + \psZero[xShift=-0.05,yShift=0.15,postString=2](2,4){\funkf}{N2} + \psZero[xShift=-0.45,yShift=0.15,postString=3](4,6){\funkf}{N3} + \psZero[xShift=-0.45,yShift=0.15,postString=4](6,7){\funkf}{N4} + \psZero[xShift=-0.45,yShift=0.15,postString=5](9,11){\funkf}{N5} + \psZero[xShift=-1.15,yShift=0,PointName=M, + postString={m=1}](0.5,2){Derive(1,\funkf)-1+\funkf}[\funkf]{M}% +} +\pcline{->}(0.5,-1)(M) +\nbput[nrot=:U,labelsep=0.01]{% + \scriptsize Steigung ist hier + \psPrintValueNew[PSfont=Palatino-Roman,decimals=0,round,fontscale=7]{nMx,{Derive(1,\funkf)}}} +\psdot[linecolor=green,strokeopacity=0.8](*{nMx} {\funkf}) +\uput[90](*{nMx} {\funkf}){$m=$ + \psPrintValueNew[PSfont=Palatino-Roman,decimals=0,round,fontscale=8]{nMx,{Derive(1,\funkf)}}} +\end{pspicture} +\end{lstlisting} +%\end{LTXexample} + + +As an alternative the values of the zeros can be placed by using the optional arguments +\Lkeyword{labelangle} and +\Lkeyword{labeldistance}: + + +\begin{LTXexample}[pos=t] +\definecolor{BeigeTS}{rgb}{0.98,0.95,0.87} +\definecolor{CornBlauTS}{rgb}{0.39,0.59,0.93} +\definecolor{SandBraun}{rgb}{0.96,0.64,0.38} +\psset{yunit=1.25cm,arrowinset=0.02,arrowlength=2,linewidth=0.5pt,saveNodeCoors,NodeCoorPrefix=n,comma} +\def\funkf{2*sqrt(x)*cos(ln(x))*sin(x)} +\begin{pspicture}[plotpoints=500,algebraic,fontscale=5,markZeros, + PointName=N,dotscale=0.7](-0.5,-3)(10,2.5) +\psStep[fillstyle=solid,fillcolor=BeigeTS,opacity=0.7,linewidth=0.3pt, + linecolor=SandBraun!50](0.001,9.5){40}{\funkf} +\psStep[StepType=Riemann,fillstyle=solid,opacity=0.3,fillcolor=CornBlauTS, + linecolor=CornBlauTS,linewidth=0.3pt](0.001,9.5){40}{\funkf} +\psaxes[labelFontSize=\scriptstyle,ticksize=-0.1 0]{->}(0,0)(0,-2.75)(10,2.5) +\psplot[linecolor=BeigeTS!60,linewidth=0.8pt]{0.001}{9.75}{\funkf} +\psplotTangent[linecolor=blue,Derive={Derive(1,\funkf)}]{1.29}{1.5}{\funkf} +\uput[90](6,1.2){$f(x)=2\cdot\sqrt{x}\cdot\cos{(\ln{x})}\cdot\sin{x}$} +{\psset{dotscale=1.5,linecolor=blue!50!black!90,ydecimals=0,Framed,opacity=0.8,decimals=1,PrintCoord} + \psZero[labelangle=-90,labeldistance=0.3,postString=1,Newton](0.5,1){\funkf}{N1} + \psZero[labelangle=-90,labeldistance=0.3,postString=2](2,4){\funkf}{N2} + \psZero[labelangle=-90,labeldistance=0.3,postString=3](4,6){\funkf}{N3} + \psZero[labelangle=-90,labeldistance=0.3,postString=4](6,7){\funkf}{N4} + \psZero[labelangle=-90,labeldistance=0.3,PointName=x,postString=5,xory,PrintCoord=false, + linestyle=none,fillcolor=green,opacity=0.6](9,11){\funkf}{N5} + \psZero[labelangle=-90,labeldistance=0.3,PointName=M,decimals=0,linestyle=none,fillcolor=SandBraun, + ydecimals=1,opacity=0.8,postString={m=1}](0.5,2){Derive(1,\funkf)-1+\funkf}[\funkf]{M}% +} +\pcline{->}(0.5,-1)(M) +\nbput[nrot=:U,labelsep=0.3,npos=0.2]{% + \scriptsize \psZero[originV=true,xory=true,onlyYVal=true,PointName=f(x),postString={m=1},Framed, + opacity=0.8,linestyle=none,markZeros=false,fontscale=10](0.5,2){Derive(1,\funkf)-1+\funkf}[\funkf]{R}} +\psdot[linecolor=green,strokeopacity=0.8](M) +\uput{0.5}[40](M){\psZero[originV=true,approx=false,xory=true,onlyYVal=true, + PointName=m,postString={m=1},markZeros=false,fontscale=8](0.5,2){Derive(1,\funkf)-1}[1]{R}} +\end{pspicture} +\end{LTXexample} + + + +\psset{unit=1cm} + + +\clearpage +\section{\Lcs{psFourier}} +A Fourier sum has the form: +% +\begin{align} +s(x) = \frac{a_0}{2} & + a_1\cos{\omega x} + a_2\cos{2\omega x} + + a_3\cos{3\omega x} + + \ldots + a_n\cos{n\omega x}\\ + & + b_1\sin{\omega x} + b_2\sin{2\omega x} + b_3\sin{3\omega x} + + \ldots + b_m\sin{m\omega x} +\end{align} +% +\noindent The macro \Lcs{psFourier} plots \Index{Fourier sums}. The +syntax is similiar to \Lcs{psPolynomial}, except that there are +two kinds of coefficients: + +\begin{BDef} +\Lcs{psFourier}\OptArgs\Largb{xStart}\Largb{xEnd} +\end{BDef} + +The coefficients must have the orders $cosCoeff=a_0\ a_1\ a_2\ \ldots$ +and $sinCoeff=b_1\ b_2\ b_3\ \ldots$ and be separated by +\textbf{spaces}. The default is \Lkeyword{cosCoeff}=0,\Lkeyword{sinCoeff}=1, +which gives the standard \verb+sin+ function. Note that +%%JF, I think it is better without the angle brackets, but +%%you know the conventions used better than I do, so you +%%may disagree. +%the constant value can only be set with \verb+cosCoeff=<a0>+. +the constant value can only be set with \Lkeyword{cosCoeff}=\verb+a0+. + +\begin{LTXexample} +\begin{pspicture}(-5,-3)(5,5.5) +\psaxes{->}(0,0)(-5,-2)(5,4.5) +\psset{plotpoints=500,linewidth=1pt} +\psFourier[cosCoeff=2, linecolor=green]{-4.5}{4.5} +\psFourier[cosCoeff=0 0 2, linecolor=magenta]{-4.5}{4.5} +\psFourier[cosCoeff=2 0 2, linecolor=red]{-4.5}{4.5} +\end{pspicture} +\end{LTXexample} + +\begin{LTXexample} +\psset{yunit=0.75} +\begin{pspicture}(-5,-6)(5,7) +\psaxes{->}(0,0)(-5,-6)(5,7) +\psset{plotpoints=500} +\psFourier[linecolor=red,linewidth=1pt]{-4.5}{4.5} +\psFourier[sinCoeff= -1 1 -1 1 -1 1 -1 1,% + linecolor=blue,linewidth=1.5pt]{-4.5}{4.5} +\end{pspicture} +\end{LTXexample} + +\begin{LTXexample} +\begin{pspicture}(-5,-5)(5,5.5) +\psaxes{->}(0,0)(-5,-5)(5,5) +\psset{plotpoints=500,linewidth=1.5pt} +\psFourier[sinCoeff=-.5 1 1 1 1 ,cosCoeff=-.5 1 1 1 1 1,% + linecolor=blue]{-4.5}{4.5} +\end{pspicture} +\end{LTXexample} + +\clearpage +\section{\Lcs{psBessel}} +The Bessel function of order $n$ is defined as +% +\begin{align} +J_n(x) &=\frac{1}{\pi}\int_0^\pi\cos(x\sin t-nt)\dt\\ + &=\sum_{k=0}^{\infty}\frac{(-1)^k \left(\frac{x}{2}\right)^{n+2k}}{k!\Gamma(n+k+1)} +\end{align} +% +\noindent The syntax of the macro is + +\begin{BDef} +\Lcs{psBessel}\OptArgs\Largb{order}\Largb{xStart}\Largb{xEnd} +\end{BDef} + +There are two special parameters for the Bessel function, and also the +settings of many \LPack{pst-plot} or \LPack{pstricks} parameters +affect the plot. +These two ``constants'' have the following meaning: +% +\[ +f(t) = constI \cdot J_n + constII +\] +% +\noindent +where \Lkeyword{constI} and \Lkeyword{constII} must be real PostScript expressions, e.g. + +\begin{lstlisting}[style=syntax] +\psset{constI=2.3,constII=t k sin 1.2 mul 0.37 add} +\end{lstlisting} + +The Bessel function is plotted with the parametricplot macro, this is the +reason why the variable is named \verb+t+. The internal procedure \verb+k+ +converts the value t from radian into degrees. The above setting is the same as +% +\[ +f(t) = 2.3 \cdot J_n + 1.2\cdot \sin t + 0.37 +\] +% +In particular, note that the default for +\Lkeyword{plotpoints} is 500. If the plotting computations are too +time consuming at this setting, it can be decreased in the usual +way, at the cost of some reduction in graphics resolution. + +\begin{LTXexample} +{ +\psset{xunit=0.25,yunit=5} +\begin{pspicture}(-13,-.85)(13,1.25) +\rput(13,0.8){% + $\displaystyle J_n(x)=\frac{1}{\pi}\int_0^\pi\cos(x\sin t-nt)\dt$% +} +\psaxes[Dy=0.2,Dx=4]{->}(0,0)(-30,-.8)(30,1.2) +\psset{linewidth=1pt} +\psBessel[linecolor=red]{0}{-28}{28}% +\psBessel[linecolor=blue]{1}{-28}{28}% +\psBessel[linecolor=green]{2}{-28}{28}% +\psBessel[linecolor=magenta]{3}{-28}{28}% +\end{pspicture} +} +\end{LTXexample} + +\begin{LTXexample} +{ +\psset{xunit=0.25,yunit=2.5} +\begin{pspicture}(-13,-1.5)(13,3) +\rput(13,0.8){% + $\displaystyle f(t) = 2.3 \cdot J_0 + 1.2\cdot \sin t + 0.37$% +} +\psaxes[Dy=0.8,dy=2cm,Dx=4]{->}(0,0)(-30,-1.5)(30,3) +\psset{linewidth=1pt} +\psBessel[linecolor=red,constI=2.3,constII={t k sin 1.2 mul 0.37 add}]{0}{-28}{28}% +\end{pspicture} +} +\end{LTXexample} + +\clearpage + +\clearpage +\section{Modfied Bessel function of first order} +The modified Bessel function of first order is defined as +% +\begin{align} +I_\nu(x) &= \left(\frac12 x\right)^\nu + \sum\limits_{k=0}^{\infty} \frac{{\left(\frac14 x^2\right)}^k}{k!\Gamma(\nu+k+1)} +\end{align} +% +\noindent The syntax of the macro is + +\begin{BDef} +\Lcs{psModBessel}\OptArgs\Largb{xStart}\Largb{xEnd} +\end{BDef} + +The only valid optional argument for the function is \Lkeyword{nue}, which +is preset to 0, it shows $I_0$. + +\begin{LTXexample} +\begin{pspicture}(0,-0.5)(5,5) +\psaxes[ticksize=-5pt 0]{->}(5,5) +\psModBessel[yMaxValue=5,nue=0,linecolor=red]{0}{5} +\psModBessel[yMaxValue=5,nue=1,linecolor=green]{0}{5} +\psModBessel[yMaxValue=5,nue=2,linecolor=blue]{0}{5} +\psModBessel[yMaxValue=5,nue=3,linecolor=cyan]{0}{5} +\end{pspicture} +\end{LTXexample} + +\clearpage +\section{\Lcs{psSi}, \Lcs{pssi} and \Lcs{psCi}} +The integral sin and cosin are defined as +% +\begin{align} +\mathrm{Si}(x) &= \int_0^x\dfrac{\sin t}{t}\dt\\ +\mathrm{si}(x) &= - \int_x^{\infty}\dfrac{\sin t}{t}\dt=\mathrm{Si}(x)-\frac{\pi}{2}\\ +\mathrm{Ci}(x) &= -\int_x^{\infty}\dfrac{\cos t}{t}\dt=\gamma+\ln x +\int_0^{x}\dfrac{\cos t -1}{t}\dt +\end{align} +% +\noindent The syntax of the macros is + +\begin{BDef} +\Lcs{psSi}\OptArgs\Largb{xStart}\Largb{xEnd}\\ +\Lcs{pssi}\OptArgs\Largb{xStart}\Largb{xEnd}\\ +\Lcs{psCi}\OptArgs\Largb{xStart}\Largb{xEnd} +\end{BDef} + +\begin{LTXexample}[pos=t] +\psset{xunit=0.5} +\begin{pspicture}(-15,-4.5)(15,2) + \psaxes[dx=1cm,Dx=2]{->}(0,0)(-14.1,-4)(14,2) + \psplot[plotpoints=1000]{-12.5}{12.5}{ x RadtoDeg sin x div } + \psSi[plotpoints=1500,linecolor=red,linewidth=1pt]{-13.5}{13.5} + \pssi[plotpoints=1500,linecolor=blue,linewidth=1pt]{-13.5}{13.5} + \rput(-5,1.5){\color{red}$Si(x)=\int\limits_{0}^x \frac{\sin(t)}{t}\dt$} + \rput(8,-1.5){\color{blue}$si(x)=-\int\limits_{x}^{\infty} \frac{\sin(t)}{t}\dt=Si(x)-\frac{\pi}{2}$} + \rput(8,.5){$f(x)= \frac{\sin(t)}{t}$} +\end{pspicture} +\end{LTXexample} + + + +\begin{LTXexample}[pos=t] +\psset{xunit=0.5} +\begin{pspicture*}(-13,-4.2)(13,4.2) + \psaxes[dx=1cm,Dx=2]{->}(0,0)(-12.1,-4)(12,4) + \psplot[plotpoints=1000]{-14.5}{14.5}{ x RadtoDeg cos x Div } + \psCi[plotpoints=500,linecolor=red,linewidth=1pt]{-11.5}{11.5} + \psci[plotpoints=500,linecolor=blue,linewidth=1pt]{-11.5}{11.5} + \rput(-8,1.5){\color{red}$Ci(x)=-\int\limits_{x}^{\infty} \frac{\cos(t)}{t}\dt$} + \rput(8,1.5){\color{blue}$ci(x)=-Ci(x)+\ln(x)+\gamma$} +\end{pspicture*} +\end{LTXexample} + +\clearpage +\section{\nxLcs{psIntegral}, \nxLcs{psCumIntegral}, and \nxLcs{psConv}} +These new macros\footnote{Created by Jose-Emilio Vila-Forcen} +allows to plot the result of an integral using the Simpson numerical integration rule. +The first one is the result of the integral of a function with two variables, and +the integral is performed over one of them. The second one is the cumulative +integral of a function (similar to \Lcs{psGaussI} but valid for all functions). +The third one is the result of a convolution. They are defined as: +% +\begin{align} +\text{\Lcs{psIntegral}}(x) &= \int\limits_a^b f(x,t)\mathrm{d}t \\ +\text{\Lcs{psCumIntegral}}(x) &= \int\limits_{\text{xStart}}^{x} f(t)\mathrm{d}t \\ +\text{\Lcs{psConv}}(x) &= \int\limits_a^b f(t)g(x-t)\mathrm{d}t +\end{align} +% +In the first one, the integral is performed from $a$ to $b$ and the function $f$ depends +on two parameters. In the second one, the function $f$ depends on only one parameter, and the +integral is performed from the minimum value specified for $x$ (\Lkeyword{xStart}) and the current +value of $x$ in the plot. The third one uses the \Lcs{psIntegral} macro to perform an approximation +to the convolution, where the integration is performed from $a$ to $b$. + +The syntax of these macros is: + +\begin{BDef} +\Lcs{psIntegral}\OptArgs\Largb{xStart}\Largb{xEnd}\Largr{a,b}\Largb{ function }\\ +\Lcs{psCumIngegral}\OptArgs\Largb{xStart}\Largb{xEnd}\Largb{ function }\\ +\Lcs{psConv}\OptArgs\Largb{xStart}\Largb{xEnd}\Largr{a,b}\Largb{ function f }\Largb{ function g } +\end{BDef} + +In the first macro, the function should be created such that it accepts two values: \verb|<x t function>| +should be a value. For the second and the third functions, they only need to accept one +parameter: \verb|<x function>| should be a value. + +There are no new parameters for these functions. The two most important ones are \Lkeyword{plotpoints}, +which controls the number of points of the plot (number of divisions on $x$ for the plot) and +\Lkeyword{Simpson}, which controls the precision of the integration (a larger number means a smallest +step). The precision and the smoothness of the plot depend strongly on these two parameters. + +\bigskip +\begin{LTXexample} +%\usepackage{pst-math} +\psset{xunit=0.5cm,yunit=2cm} +\begin{pspicture}[linewidth=1pt](-10,-.5)(10,1.5) + \psaxes[dx=1cm,Dx=2]{->}(0,0)(-10,0)(10,1.5) + \psCumIntegral[plotpoints=200,Simpson=10]{-10}{10}{0 1 GAUSS} + \psIntegral[plotpoints=200,Simpson=100,linecolor=green]{.1}{10}(-3,3){0 exch GAUSS} + \psIntegral[plotpoints=200,Simpson=10,linecolor=red, + fillcolor=red!40,fillstyle=solid,opacity=0.5]{-10}{10}(-4,6){1 GAUSS} +\end{pspicture} +\end{LTXexample} + +In the example, the cumulative integral of a Gaussian is presented in black. In red, a +Gaussian is varying its mean from -10 to 10, and the result is the integral from -4 to 6. +Finally, in green it is presented the integral of a Gaussian from -3 to 3, where the +variance is varying from 0.1 to 10. + + +\psset{algebraic=false} + +\begin{LTXexample} +\psset{xunit=1cm,yunit=4cm} +\begin{pspicture}[linewidth=1pt](-5,-.2)(5,0.75) + \psaxes[dx=1cm,Dx=1,Dy=0.5]{->}(0,0)(-5,0)(5,0.75) + \psplot[linecolor=blue,plotpoints=200]{-5}{5}{x abs 2 le {0.25}{0} ifelse} + \psplot[linecolor=green,plotpoints=200]{-5}{5}{x abs 1 le {.5}{0} ifelse} + \psConv[plotpoints=100,Simpson=1000,linecolor=red]{-5}{5}(-10,10)% + {abs 2 le {0.25}{0} ifelse}{abs 1 le {.5} {0} ifelse} +\end{pspicture} +\end{LTXexample} + +In the second example, a convolution is performed using two rectangle functions. +The result (in red) is a \Index{trapezoid function}. + + +\begin{LTXexample} +\psset{xunit=0.5cm,yunit=4cm} +\begin{pspicture}[linewidth=1pt](-11,-1.5)(11,1.5) + \psaxes[dx=1cm,Dx=2]{->}(0,0)(-10.5,-1.25)(10.5,1.25) + \psCumIntegral[plotpoints=2000,Simpson=10,algebraic]{-10}{10}{-sin(x/2)/2} + \psplot[plotpoints=2000,linestyle=dashed,linecolor=red,algebraic]{-10}{10}{cos(x/2)} + \rput(4,0.5){\textcolor{red}{$\displaystyle\cos\left(\frac{x}2\right)$}} + \rput*(0,-1.1){$\displaystyle\int\limits\frac{-\sin(\frac{x}2)}{2}\mathrm dx$} +\end{pspicture} +\end{LTXexample} + + + + +\clearpage +\section{Distributions} +All distributions which use the $\Gamma$- or $\ln\Gamma$-function need the \LPack{pst-math} package, +it defines the PostScript functions \Lps{GAMMA} and \Lps{GAMMALN}. \LPack{pst-func} reads by default the PostScript +file \LFile{pst-math.pro}. It is part of any \TeX\ distribution and should also be on +your system, otherwise install or update it from \textsc{CTAN}. It must be the latest version. + +\begin{LTXexample}[pos=l,width=7cm] +\begin{pspicture*}(-0.5,-0.5)(6.2,5.2) + \psaxes{->}(0,0)(6,5) + \psset{plotpoints=100,linewidth=1pt} + \psplot[linecolor=red]{0.01}{4}{ x GAMMA } + \psplot[linecolor=blue]{0.01}{5}{ x GAMMALN } +\end{pspicture*} +\end{LTXexample} + +\clearpage +\subsection{Normal distribution (Gauss)} +The Gauss function is defined as +% +\begin{align} +f(x) &= \dfrac{1}{\sigma\sqrt{2\pi}}\,e^{-\dfrac{\left(x-\mu\right)^2}{2\sigma{}^2}} +\end{align} +% +\noindent The syntax of the macros is + +\begin{BDef} +\Lcs{psGauss}\OptArgs\Largb{xStart}\Largb{xEnd}\\ +\Lcs{psGaussI}\OptArgs\Largb{xStart}\Largb{xEnd} +\end{BDef} + +\noindent where the only new parameter are \Lkeyword{sigma}=<value>+ and \Lkeyword{mue}=<value>+ +for the horizontal shift, which can also be set in the usual way with \Lcs{psset}. +It is significant only for the \Lcs{psGauss} and \Lcs{psGaussI} macro. The default is +\Lkeyword{sigma}=0.5 and \Lkeyword{mue}=0. The integral is caclulated wuth the Simson algorithm +and has one special option, called \Lkeyword{Simpson}, which defines the number of intervals per step +and is predefined with 5. + +\begin{LTXexample}[pos=t,preset=\centering,wide=true] +\psset{yunit=4cm,xunit=3} +\begin{pspicture}(-2,-0.2)(2,1.4) +% \psgrid[griddots=10,gridlabels=0pt, subgriddiv=0] + \psaxes[Dy=0.25]{->}(0,0)(-2,0)(2,1.25) + \uput[-90](6,0){x}\uput[0](0,1){y} + \rput[lb](1,0.75){\textcolor{red}{$\sigma =0.5$}} + \rput[lb](1,0.5){\textcolor{blue}{$\sigma =1$}} + \rput[lb](-2,0.5){$f(x)=\dfrac{1}{\sigma\sqrt{2\pi}}\,e^{-\dfrac{(x-\mu)^2}{2\sigma{}^2}}$} + \psGauss[linecolor=red, linewidth=2pt]{-1.75}{1.75}% + \psGaussI[linewidth=1pt]{-2}{2}% + \psGauss[linecolor=cyan, mue=0.5, linewidth=2pt]{-1.75}{1.75}% + \psGauss[sigma=1, linecolor=blue, linewidth=2pt]{-1.75}{1.75} +\end{pspicture} +\end{LTXexample} + + + + + +\clearpage + + + +\subsection{Binomial distribution}\label{sec:bindistri} +\begin{sloppypar} +The following five macros plot binomial probability mass function \Lcs{psBinomial} and \Lcs{psBinomialC} in curve style, the normalized one is \Lcs{psBinomialN}. The cumulative distribution function $F$ \Lcs{psBinomialF} and the complement of the cumulative distribution function ($1-F$) \Lcs{psBinomialFS} +The vertical range for the plots is the $y$-Intervall $[0;1]$. +Rescaling other values can be done by setting the \Lkeyword{yunit} option +to any other value. +\end{sloppypar} + +The binomial distribution \Lcs{psBinomial} gives the discrete probability distribution $P_p(n|N)$ $n$ successes out of $N$ Bernoulli trials (where the result of each Bernoulli trial is true with probability $p$ and false with probability $q=1-p$). The binomial distribution is therefore given by + +\begin{align} +P_p(n|N) &= \binom{N}{n}p^nq^{N-n} \\ + &= \frac{N!}{n!(N-n)!}p^n(1-p)^{N-n}, +\end{align} + +where $(N; n)$ is a binomial coefficient and $P$ the probability. + +The syntax is: + +\begin{BDef} +\Lcs{psBinomial}\OptArgs\Largb{N}\Largb{probability p}\\ +\Lcs{psBinomial}\OptArgs\Largb{m,N}\Largb{probability p}\\ +\Lcs{psBinomial}\OptArgs\Largb{m,n,N}\Largb{probability p}\\ +\Lcs{psBinomialC}\OptArgs\Largb{N}\Largb{probability p}\\ +\Lcs{psBinomialN}\OptArgs\Largb{N}\Largb{probability p}\\ +\Lcs{psBinomialF}\OptArgs\Largb{N}\Largb{probability p}\\ +\Lcs{psBinomialF}\OptArgs\Largb{m,N}\Largb{probability p}\\ +\Lcs{psBinomialF}\OptArgs\Largb{m,n,N}\Largb{probability p}\\ +\Lcs{psBinomialFS}\OptArgs\Largb{N}\Largb{probability p}\\ +\Lcs{psBinomialFS}\OptArgs\Largb{m,N}\Largb{probability p}\\ +\Lcs{psBinomialFS}\OptArgs\Largb{m,n,N}\Largb{probability p} +\end{BDef} + +\begin{itemize} +\item with one argument $N$ the sequence $0\ldots N$ is calculated and plotted +\item with two arguments $m,N$ the sequence $0\ldots N$ is calculated and + the sequence $m\ldots N$ is plotted +\item with three arguments $m,n,N$ the sequence $0\ldots N$ is calculated and + the sequence $m\ldots n$ is plotted +\end{itemize} + +Now \Lcs{psBinomial}, \Lcs{psBinomialF} and \Lcs{psBinomialFS} uses a new code, so the old restriction in using the value for $N$ (old: $N<100$) is no longer valid. A new limit vor $N$ is not searched and it's not found. +The valid options for the macros are \Lkeyword{markZeros} to draw rectangles instead +of a continous line and \Lkeyword{printValue} for printing the $y$-values in the color LabelColor $=$ color on top of the lines in distance labelsep and xlabelsep, rotated by labelangle $=\alpha$. For this option all other options from section~1 +for the macro \Lcs{psPrintValue} are valid, too.~ \cite{pst-tools} Important is the keyword \Lkeyword{valuewidth} +which is preset to 15. If your value has more characters when converting into a string, it will +not be printed or cause an GhostScript error. + +Special options are +\begin{itemize} +\item \Lkeyword{barwidth}, which is a factor (no dimension) and set by default to 1. This option is not valid for +the macro \Lcs{psBinomialN}! +\item \Lkeyword{alternateColors} is a new fillstyle, so the colors alternates between \Lkeyword{fillcolorA} and \Lkeyword{fillcolorB}, only valid for \Lcs{psBinomial}. +\item \Lkeyword{fillcolorA} alternate color one. +\item \Lkeyword{fillcolorB} alternate color two. +\item \Lkeyword{labelangle} is the rotation of the printed values, default is 90\textdegree +\item \Lkeyword{xlabelsep} is the x-separation of the printed values, default is 0 (no dimension) +\item \Lkeyword{labelsep} is the y-separation of the printed values, default is 0.2 (no dimension) +\item \Lkeyword{LabelColor} is the color of the printed values, default is black +\item \Lkeyword{PrintVLimit} is the value limit for the printed values, default is $1e-64$, smaller values are not printed. +\item \Lkeyword{Switch2Log} is the value for $N$ where the new calculation is used, default is $80$. +\item \Lkeyword{LineEnding} this boolean is only valid for the macros \Lcs{psBinomialF} and \Lcs{psBinomialFS}, default is true. Draws circles at the end of the lines +\item \Lkeyword{VLines} this option is only valid for the macros \Lcs{psBinomialF} and \Lcs{psBinomialFS}, default is false. Draws the vertical lines dashed. +\item \Lkeyword{rightEnd}, this option is only valid for the macros \Lcs{psBinomialF} and \Lcs{psBinomialFS} when LineEnding=true (default) and $n=N$, default is 2 +\item \Lkeyword{leftEnd}, this option is only valid for the macros \Lcs{psBinomialF} and \Lcs{psBinomialFS} when LineEnding=true (default) and $m=0$, default is 1 +\item \Lkeyword{radiusout}, this option is only valid for the macros \Lcs{psBinomialF} and \Lcs{psBinomialFS} when LineEnding=true (default) for the outer radius of the both dots left and right, default is 2 +\item \Lkeyword{radiusinL}, this option is only valid for the macros \Lcs{psBinomialF} and \Lcs{psBinomialFS} when LineEnding=true (default) for the inner radius of the left dot, default is 0 +\item \Lkeyword{radiusinR}, this option is only valid for the macros \Lcs{psBinomialF} and \Lcs{psBinomialFS} when LineEnding=true (default) for the inner radius of the right dot, default is 1.5 +\item \Lkeyword{LineEndColorL} this option is only valid for the macros \Lcs{psBinomialF} and \Lcs{psBinomialFS} when LineEnding=true (default) for the color of the left dot, default is green +\item \Lkeyword{LineEndColorR} this option is only valid for the macros \Lcs{psBinomialF} and \Lcs{psBinomialFS} when LineEnding=true (default) for the inner radius of the right dot, default is red +\item \Lkeyword{LeftClipX} gives the left end of the clipping area for \Lcs{psBinomialC}, default is $-1$. +\item \Lkeyword{RightClipX} gives the distance to $N$ for the right end of the clipping area for \Lcs{psBinomialC}, default is $1$. +\end{itemize} + + +\psset[pst-func]{barwidth=1} +\begin{LTXexample}[pos=t,preset=\centering] +\psset{xunit=1cm,yunit=5cm}% +\begin{pspicture}(-1,-0.15)(7,0.6)% +\psaxes[Dy=0.2,dy=0.2\psyunit]{->}(0,0)(-1,0)(7,0.5) +\uput[-90](7,0){$k$} \uput[90](0,0.5){$P(X=k)$} +\psBinomial[markZeros,printValue,fillstyle=vlines, +labelangle=80,LabelColor=blue]{6}{0.4} +\end{pspicture} +\end{LTXexample} + +\begin{LTXexample}[pos=t,preset=\centering] +\psset{xunit=1cm,yunit=10cm}% +\begin{pspicture}(-1,-0.05)(8,0.6)% +\psaxes[Dy=0.2,dy=0.2\psyunit]{->}(0,0)(-1,0)(8,0.5) +\uput[-90](8,0){$k$} \uput[90](0,0.5){$P(X=k)$} +\psBinomialC[fillstyle=solid,opacity=0.5,fillcolor=cyan]{7}{0.6} +\psBinomial[linecolor=red,markZeros,printValue,fillstyle=solid, + fillcolor=blue,barwidth=0.2,xlabelsep=-0.05]{7}{0.6} +\end{pspicture} +\end{LTXexample} + +\begin{LTXexample}[pos=t,preset=\centering] +\psset{xunit=1cm,yunit=10cm}% +\begin{pspicture}(-1,-0.05)(8,0.6)% +\psaxes[Dy=0.2,dy=0.2\psyunit]{->}(0,0)(-1,0)(8,0.5) +\uput[-90](8,0){$k$} \uput[90](0,0.5){$P(X=k)$} +\psBinomial[linecolor=black!30]{0,7}{0.6} +\psBinomial[linecolor=blue,markZeros,printValue,fillstyle=solid, + fillcolor=blue,barwidth=0.4]{2,5,7}{0.6} +\psBinomialC[showpoints=true]{7}{0.6} +\end{pspicture} +\end{LTXexample} + +\begin{LTXexample}[pos=t,preset=\centering] +\psset{xunit=0.25cm,yunit=10cm} +\begin{pspicture*}(-1,-0.05)(61,0.52) +\psaxes[Dx=5,dx=5\psxunit,Dy=0.2,dy=0.2\psyunit]{->}(60,0.5) +\uput[-90](60,0){$k$} \uput[0](0,0.5){$P(X=k)$} +\psBinomial[markZeros,linecolor=red]{4}{.5} +\psset{linewidth=1pt} +\psBinomial[linecolor=green]{5}{.5} \psBinomial[linecolor=blue]{10}{.5} +\psBinomial[linecolor=red]{20}{.5} \psBinomial[linecolor=magenta]{50}{.5} +\psBinomial[linecolor=cyan]{0,55,75}{.5} +\end{pspicture*} +\end{LTXexample} + +\begin{LTXexample}[pos=t,preset=\centering] +\psset{xunit=0.8cm,yunit=8cm}% +\begin{pspicture*}[showgrid=false](-1.5,-0.1)(16,1.2)% +\psset{arrowscale=1.3,arrowinset=0.05,arrowlength=1.9,comma}% +\psaxes[labelFontSize=\scriptstyle,xticksize=0 1.07,yticksize=0 16,tickcolor=gray!50, + Dy=0.1,dy=0.1,Dx=1,dx=1,Ox=0]{->}(0,0)(-0.9,0)(16,1.1) +\uput[-90](15.8,0){$z$}\uput[0](0,1.1){$P_{0,15}^{100}(Z=z)$} +\psBinomialC[linecolor=cyan,fillstyle=solid,fillcolor=cyan!50,opacity=0.4]{40}{0.15}% +\psBinomial[markZeros,linecolor=BrickRed,fillstyle=solid,fillcolor=BrickRed,barwidth=0.75,opacity=0.6]{1,16,40}{0.15}% +\psBinomialFS[markZeros,linecolor=Green,fillstyle=solid,fillcolor=orange,barwidth=0.3,opacity=0.6]{0,16,40}{0.15}% +\psBinomialF[linecolor=gray,fillstyle=solid,fillcolor=yellow,barwidth=0.4,opacity=0.5]{3,16,40}{0.15} +\end{pspicture*} +\end{LTXexample} + +\begin{LTXexample}[pos=t,preset=\centering] +\psset{xunit=0.75cm,yunit=7.5cm}% +\begin{pspicture*}[showgrid=false](-1.3,-0.067)(14.67,1.13)% +\psset{arrowscale=1.3,arrowinset=0.05,arrowlength=1.9,comma} +\psaxes[labelFontSize=\scriptstyle,xticksize=0 1.07,yticksize=0 12,tickcolor=gray!50,Dy=0.1,dy=0.1,Dx=1,dx=1,Ox=0]{->}(0,0)(-0.9,0)(14,1.1) +\uput[-90](13.8,0){$z$} \uput[0](0,1.08){$F_{0,7}^{10}(Z\leq z)$} +\psBinomial[markZeros,linecolor=orange,fillstyle=solid,fillcolor=orange,barwidth=1,opacity=0.5]{0,10,10}{0.7} +\psBinomialF[markZeros,linecolor=blue,linewidth=0.7pt,barwidth=0.2, +opacity=0.5,fillstyle=solid,fillcolor=blue,valuewidth=15]{0,13,10}{0.7} +\psBinomialFS[LineEnding=false,linecolor=BrickRed,linewidth=0.9pt,VLines=true]{0,10,10}{0.7} +\psBinomialF[linecolor=Green,printValue=false,linewidth=1.2pt,LineEndColorR=BrickRed,LineEndColorL=Green!70, +radiusout=3.5,radiusinL=0,radiusinR=2,LineEnding=true,leftEnd=1,rightEnd=3]{0,10,10}{0.7} +\end{pspicture*} +\end{LTXexample} + +\begin{LTXexample}[pos=t,preset=\centering] +\begin{pspicture}[showgrid=false](-.75,-1.8)(13.2,4.7)% +{\psset{xunit=1cm,yunit=12cm}% +\psset{plotpoints=500,arrowscale=1.3,arrowinset=0.05,arrowlength=1.9,comma} +\psaxes[labelFontSize=\scriptstyle,xticksize=0 0,yticksize=0 12,tickcolor=gray!50,Dy=0.05,dy=0.05,Dx=1,dx=1,Ox=0]{-}(0,0)(-0.9,0)(10.8,0.34) +\uput[-90](11.9,0){$z$} \uput[0](0,0.36){$P_{0,8}^{10}(Z=z)$}\uput[0](0,0.32){$P_{0,7}^{10}(Z=z)$} +\rput(-0.05,0){% +\psBinomialC[linecolor=Green,fillstyle=solid,fillcolor=gray,opacity=0.25,plotstyle=curve,linestyle=dashed]{10}{0.8}} +\rput(0.05,0){% +\psBinomialC[linecolor=cyan,fillstyle=solid,fillcolor=cyan,opacity=0.25,plotstyle=curve,linestyle=dashed]{10}{0.7}% +\psBinomial[markZeros,linecolor=cyan,fillstyle=solid,fillcolor=cyan,barwidth=0.2,opacity=0.85]{0,8,10}{0.7}%,printValue +\psBinomial[markZeros,linecolor=magenta,fillstyle=solid,fillcolor=magenta,barwidth=0.2,opacity=0.85]{9,10,10}{0.7} +} +\rput(-0.05,0){% +\psBinomialC[linecolor=Green,fillstyle=solid,fillcolor=gray,opacity=0.25,plotstyle=curve,linestyle=dashed]{10}{0.8} +\psBinomial[markZeros,linecolor=DeepSkyBlue4,fillstyle=solid,fillcolor=DeepSkyBlue4,barwidth=0.2,opacity=0.85]{0,8,10}{0.8}%,printValue +\psBinomial[markZeros,linecolor=BrickRed,fillstyle=solid,fillcolor=BrickRed,barwidth=0.2,opacity=0.85]{9,10,10}{0.8} +} +\psaxes[labels=none,xticksize=-2pt 0,yticksize=-2pt 0,tickcolor=black!70,Dy=0.05,dy=0.05\psyunit,Dx=1,dx=1\psxunit,Ox=0]{->}(0,0)(-0.9,0)(12,0.35) +\rput(5,0.33){\psframebox[fillstyle=solid,fillcolor=orange!30,linestyle=none]{$n=10$}} +} +\end{pspicture} +\end{LTXexample} + +\begin{LTXexample}[pos=t,preset=\centering] +\begin{pspicture}[showgrid=false](-.75,-1.8)(13.2,4.7)% +{\psset{xunit=1cm,yunit=12cm}% +\psset{plotpoints=500,arrowscale=1.3,arrowinset=0.05,arrowlength=1.9,comma} +\psaxes[labelFontSize=\scriptstyle,xticksize=0 0,yticksize=0 12,tickcolor=gray!50,Dy=0.05,dy=0.05,Dx=1,dx=1,Ox=0]{-}(0,0)(-0.9,0)(10.8,0.34) +\uput[-90](11.9,0){$z$} \uput[0](0,0.32){$P_{0,7}^{10}(Z=z)$} +\psBinomialC[linecolor=cyan,fillstyle=solid,fillcolor=cyan,opacity=0.25,plotstyle=curve,linestyle=dashed,LeftClipX=4,RightClipX=-3]{10}{0.7}% +\psBinomial[markZeros,linecolor=cyan,fillstyle=solid,fillcolor=cyan,barwidth=0.2,opacity=0.85]{0,8,10}{0.7}%,printValue +\psBinomial[markZeros,linecolor=magenta,fillstyle=solid,fillcolor=magenta,barwidth=0.2,opacity=0.85]{9,10,10}{0.7} +\psaxes[labels=none,xticksize=-2pt 0,yticksize=-2pt 0,tickcolor=black!70,Dy=0.05,dy=0.05\psyunit,Dx=1,dx=1\psxunit,Ox=0]{->}(0,0)(-0.9,0)(12,0.35) +\rput(5,0.33){\psframebox[fillstyle=solid,fillcolor=orange!30,linestyle=none]{$n=10$}} +} +\end{pspicture} +\end{LTXexample} + +\begin{LTXexample}[pos=t,preset=\centering] +{\psset{xunit=0.07cm,yunit=10cm}% +\begin{pspicture}[showgrid=false](-6,-0.1)(220,1.1)% +\psset{plotpoints=500,arrowscale=1.3,arrowinset=0.05,arrowlength=1.9,comma} +\psaxes[labelFontSize=\scriptstyle,xticksize=0 1,yticksize=0 218,tickcolor=gray!50,Dy=0.05,dy=0.05,Dx=10,dx=10,showorigin=false]{->}(0,0)(219,1.05) +\uput[-90](219,0){$k$} \uput[0](0,1.05){$P(X=k)=B(300;\frac{1}{3};k)$} +\psBinomial[linecolor=Green,fillstyle=solid,fillcolor=cyan,opacity=0.5,printValue=true,markZeros,fontscale=4,xlabelsep=-0.175,LabelColor=Green,labelangle=80,PrintVLimit=0.01]{1,210,300}{1 3 div}%,printValue +\psBinomialF[radiusout=1.3,radiusinR=0.9,linecolor=cyan,leftEnd=4,rightEnd=5,linewidth=0.8pt,LineEndColorR=DeepSkyBlue4,LineEndColorL=DeepSkyBlue4,VLines,printValue,fontscale=4,LabelColor=cyan]{0,230,300}{1 3 div} +\psBinomialFS[radiusout=1.3,radiusinR=0.9,linecolor=red,leftEnd=4,rightEnd=5,linewidth=0.8pt,LineEndColorR=DeepSkyBlue4,LineEndColorL=red,VLines,printValue,fontscale=4,labelangle=50,LabelColor=orange]{0,200,300}{1 3 div} +\end{pspicture}} +\end{LTXexample} + +The default binomial distribution has the mean of $\mu=E(X)=N\cdot p$ +and a variant of $\sigma^2=\mu\cdot(1-p)$. +The normalized distribution has a mean of $0$. Instead of $P(X=k)$ +we use $P(Z=z)$ with $Z=\dfrac{X-E(X)}{\sigma(X)}$ and $P\leftarrow P\cdot\sigma$. +The macros use the recursive definition of the binomial distribution: +% +\begin{align} +P(k) = P(k-1)\cdot\frac{N-k+1}{k}\cdot\frac{p}{1-p} +\end{align} + +\begin{LTXexample}[pos=t,preset=\centering] +\psset{xunit=1cm,yunit=5cm}% +\begin{pspicture}(-3,-0.15)(4,0.55)% +\psaxes[Dy=0.2,dy=0.2\psyunit]{->}(0,0)(-3,0)(4,0.5) +\uput[-90](4,0){$z$} \uput[0](0,0.5){$P(Z=z)$} +\psBinomialN[markZeros,fillstyle=vlines]{6}{0.4} +\end{pspicture} +\end{LTXexample} + +\begin{LTXexample}[pos=t,preset=\centering] +\psset{yunit=10} +\begin{pspicture*}(-8,-0.07)(8.1,0.55) +\psaxes[Dy=0.2,dy=0.2\psyunit]{->}(0,0)(-8,0)(8,0.5) +\uput[-90](8,0){$z$} \uput[0](0,0.5){$P(Z=z)$} +\psBinomialN{125}{.5} +\psBinomialN[markZeros,linewidth=1pt,linecolor=red]{4}{.5} +\end{pspicture*} +\end{LTXexample} + +\begin{LTXexample}[pos=t,preset=\centering] +\psset{yunit=10} +\begin{pspicture*}(-8,-0.07)(8.1,0.52) +\psaxes[Dy=0.2,dy=0.2\psyunit]{->}(0,0)(-8,0)(8,0.5) +\uput[-90](8,0){$z$} \uput[0](0,0.5){$P(Z=z)$} +\psBinomialN[markZeros,linecolor=red]{4}{.5} +\psset{linewidth=1pt} +\psBinomialN[linecolor=green]{5}{.5}\psBinomialN[linecolor=blue]{10}{.5} +\psBinomialN[linecolor=red]{20}{.5} \psBinomialN[linecolor=gray]{50}{.5} +\end{pspicture*} +\end{LTXexample} + +For the normalized distribution the plotstyle can be set to \Lkeyval{curve} (\Lkeyset{plotstyle=curve}), +then the binomial distribution looks like a normal distribution. This option is only +valid for \Lcs{psBinomialN}. The option \Lkeyword{showpoints} is valid if \Lkeyval{curve} was chosen. + +\begin{LTXexample}[pos=t,preset=\centering] +\psset{xunit=1cm,yunit=10cm}% +\begin{pspicture*}(-4,-0.06)(4.1,0.57)% +\psaxes[Dy=0.2,dy=0.2\psyunit]{->}(0,0)(-4,0)(4,0.5)% +\uput[-90](4,0){$z$} \uput[90](0,0.5){$P(Z=z)$}% +\psBinomialN[linecolor=red,fillstyle=vlines,showpoints=true,markZeros]{36}{0.5}% +\psBinomialN[linecolor=blue,showpoints=true,plotstyle=curve]{36}{0.5}% +\end{pspicture*} +\end{LTXexample} + +\begin{LTXexample}[pos=t,preset=\centering] +\psset{xunit=1cm,yunit=10cm}% +\begin{pspicture*}(-4,-0.06)(4.2,0.57)% +\psaxes[Dy=0.2,dy=0.2\psyunit]{->}(0,0)(-4,0)(4,0.5)% +\uput[-90](4,0){$z$} \uput[90](0,0.5){$P(Z=z)$}% +\psBinomialN[linecolor=red]{10}{0.6}% +\psBinomialN[linecolor=blue,showpoints=true,plotstyle=curve]{10}{0.6}% +\end{pspicture*} +\end{LTXexample} + + + +\begin{LTXexample}[pos=t,preset=\centering] +\definecolor{A1}{RGB}{28, 134, 238} +\definecolor{A2}{RGB}{124, 205, 124} +\psset{xunit=4mm,yunit=70cm,arrowscale=1.5}% +\begin{pspicture*}(-2,-0.01)(30,0.25) +\psBinomial[fillstyle=alternateColors, + fillcolorA=A1,fillcolorB=A2, + markZeros]{60}{0.25} +\psaxes[Dx=5,dx=5\psxunit,Dy=0.1,dy=0.1\psyunit, + arrows=D>]{->}(28,0.13)[\Large$k$,-90][\Large$P(X=k)$,0] +\end{pspicture*}% +\end{LTXexample} + + + + + + + +\clearpage +\subsection{Poisson distribution} +Given a Poisson process\footnote{\url{http://mathworld.wolfram.com/PoissonProcess.html}}, +the probability of obtaining exactly $n$ successes in $N$ trials is given by the +limit of a binomial distribution (see Section~\ref{sec:bindistri}) +% +\begin{align} +P_p(n|N) = \frac{N!}{n!(N-n)!}\cdot p^n(1-p)^{N-n}\label{eq:normaldistri} +\end{align} +% +Viewing the distribution as a function of the expected number of successes; +% +\begin{align}\label{eq:nu} +\lambda = N\cdot p +\end{align} +% +instead of the sample size $N$ for fixed $p$, equation (2) then becomes +\eqref{eq:normaldistri}; +% +\begin{align}\label{eq:nuN} +P_{\frac{\lambda}{n}}(n|N) = \frac{N!}{n!(N-n)!}{\frac{\lambda}{N}}^n {\frac{1-\lambda}{N}}^{N-n} +\end{align} +% +Viewing the distribution as a function of the expected number of successes; +% +\[ P_\lambda(X=k)=\frac{\lambda^k}{k!}\,e^{-\lambda} \]. +% +Letting the sample size become large ($N\to\infty$), the distribution then +approaches (with $p=\frac{\lambda}{n}$): +% +\begin{align} +\lim_{n\to\infty} P(X=k) &= \lim_{n\to\infty}\frac{n!}{(n-k)!\,k!} + \left(\frac{\lambda}{n}\right)^k \left(1-\frac{\lambda}{n}\right)^{n-k} \\ + &= \lim_{n\to\infty} \left(\frac{(n-k)!\cdot (n-k+1)\cdots(n-2)(n-1)n}{(n-k)!\,n^k}\right)\cdot\\ + &\qquad \left(\frac{\lambda^k}{k!}\right)\left(1-\frac{\lambda}{n}\right)^n + \left(1-\frac{\lambda}{n}\right)^{-k}\\ + &= \frac{\lambda^k}{k!}\cdot \lim_{n\to\infty} + \underbrace{\left(\frac{n}{n}\cdot \frac{n-1}{n}\cdot\frac{n-2}{n}\cdot\ldots\cdot + \frac{n-k+1}{n}\right)}_{\to 1} \cdot\\ + &\qquad \underbrace{\left(1-\frac{\lambda}{n}\right)^n}_{\to{e^{-\lambda}}} + \underbrace{\left(1-\frac{\lambda}{n}\right)^{-k}}_{\to 1}\\ + &= \lambda^k e^{\frac{-\lambda}{k!}} +\end{align} +% +which is known as the Poisson distribution and has the follwing syntax: + +\begin{BDef} +\Lcs{psPoisson}\OptArgs\Largb{N}\Largb{lambda}\\ +\Lcs{psPoisson}\OptArgs\Largb{M,N}\Largb{lambda} +\end{BDef} + +in which \texttt{M} is an optional argument with a default of 0. + +\begin{LTXexample}[pos=t,preset=\centering] +\psset{xunit=1cm,yunit=20cm}% +\begin{pspicture}(-1,-0.05)(14,0.25)% +\uput[-90](14,0){$k$} \uput[90](0,0.2){$P(X=k)$} +\psPoisson[linecolor=red,markZeros,fillstyle=solid, + fillcolor=blue!10,printValue,valuewidth=20]{13}{6} % N lambda +\psaxes[Dy=0.1,dy=0.1\psyunit]{->}(0,0)(-1,0)(14,0.2) +\end{pspicture} +\end{LTXexample} + +\begin{LTXexample}[pos=t,preset=\centering] +\psset{xunit=1cm,yunit=20cm}% +\begin{pspicture}(-1,-0.05)(14,0.25)% +\uput[-90](14,0){$k$} \uput[90](0,0.2){$P(X=k)$} +\psPoisson[linecolor=blue,markZeros,fillstyle=solid,barwidth=0.4, + fillcolor=blue!10,printValue,valuewidth=20]{10}{6} % N lambda +\psaxes[Dy=0.1,dy=0.1\psyunit]{->}(0,0)(-1,0)(11,0.2) +\end{pspicture} +\end{LTXexample} + +\begin{LTXexample}[pos=t,preset=\centering] +\psset{xunit=1cm,yunit=20cm}% +\begin{pspicture}(-1,-0.05)(14,0.25)% +\uput[-90](14,0){$k$} \uput[90](0,0.2){$P(X=k)$} +\psPoisson[printValue,valuewidth=20]{2,11}{6} % M,N lambda +\psaxes[Dy=0.1,dy=0.1\psyunit]{->}(0,0)(-1,0)(14,0.2) +\end{pspicture} +\end{LTXexample} + +\clearpage +\subsection{Gamma distribution} +A gamma distribution is a general type of statistical distribution that is related +to the beta distribution and arises naturally in processes for which the waiting +times between Poisson distributed events are relevant. Gamma distributions have +two free parameters, labeled $\alpha$ and $\beta$. It is defined as +% +\[ +f(x)=\frac{\beta(\beta x)^{\alpha-1}e^{-\beta x}}{\Gamma(\alpha)} \qquad +\text{for $x>0$ and $\alpha$, $\beta>0$} +\] +% +and has the syntax + +\begin{BDef} +\Lcs{psGammaDist}\OptArgs\Largb{x0}\Largb{x1} +\end{BDef} + +\begin{LTXexample}[pos=t,preset=\centering] +\psset{xunit=1.2cm,yunit=10cm,plotpoints=200} +\begin{pspicture*}(-0.75,-0.05)(9.5,0.6) + \psGammaDist[linewidth=1pt,linecolor=red]{0.01}{9} + \psGammaDist[linewidth=1pt,linecolor=blue,alpha=0.3,beta=0.7]{0.01}{9} + \psaxes[Dy=0.1]{->}(0,0)(9.5,.6) +\end{pspicture*} +\end{LTXexample} + +\clearpage +\subsection{$\chi^2$-distribution} +The $\chi^2$-distribution is a continuous probability distribution. It +usually arises when a $k$-dimensional vector's orthogonal components are +independent and each follow a standard normal distribution. +The length of the vector will then have a $\chi^2$-distribution. + +\iffalse +If Y_i have normal independent distributions with mean 0 and variance 1, then +chi^2=sum_(i==1)^rY_i^2 +(1) + +is distributed as chi^2 with r degrees of freedom. This makes a chi^2 distribution +a gamma distribution with theta=2 and alpha=r/2, where r is the number of degrees of freedom. + +More generally, if chi_i^2 are independently distributed according to a chi^2 +distribution with r_1, r_2, ..., r_k degrees of freedom, then +sum_(j==1)^kchi_j^2 + +is distributed according to chi^2 with r=sum_(j==1)^(k)r_j degrees of freedom. +\fi + +The $\chi^2$ with parameter $\nu$ is the same as a Gamma distribution +with $\alpha=\nu/2$ and $\beta=1/2$ and the syntax + +\begin{BDef} +\Lcs{psChiIIDist}\OptArgs\Largb{x0}\Largb{x1} +\end{BDef} + +\begin{LTXexample}[pos=t,preset=\centering] +\psset{xunit=1.2cm,yunit=10cm,plotpoints=200} +\begin{pspicture*}(-0.75,-0.05)(9.5,.65) + \multido{\rnue=0.5+0.5,\iblue=0+10}{10}{% + \psChiIIDist[linewidth=1pt,linecolor=blue!\iblue,nue=\rnue]{0.01}{9}} + \psaxes[Dy=0.1]{->}(0,0)(9.5,.6) +\end{pspicture*} +\end{LTXexample} + +\iffalse +The cumulative distribution function is +% +\begin{align*} +D_r(\chi^2) &= int_0^{\chi^2}\frac{t^{r/2-1}e^{-t/2}\mathrm{d}t}{\Gamma(1/2r)2^{r/2}\\ + &= 1-\frac{\Gamma(1/2r,1/2\chi^2)}{\Gamma(1/2r)} +\end{align*} +\fi + +\clearpage +\subsection{Student's $t$-distribution} + +A \Index{statistical distribution} published by \Index{William Gosset} in 1908 +under his pseudonym ``Student''. The $t$-distribution with parameter $\nu$ has +the \Index{density function} +% +\[ +f(x)=\frac1{\sqrt{\nu\pi}}\cdot + \frac{\Gamma[(\nu+1)/2]}{\Gamma(\nu/2)}\cdot\frac1{[1+(x^2/\nu)]^{(\nu+1)/2}} \qquad +\text{for $-\infty<x<\infty$ and $\nu>0$} +\] +% +and the following syntax + +\begin{BDef} +\Lcs{psTDist}\OptArgs\Largb{x0}\Largb{x1} +\end{BDef} + +\begin{LTXexample}[pos=t,preset=\centering] +\psset{xunit=1.25cm,yunit=10cm} +\begin{pspicture}(-6,-0.1)(6,.5) + \psaxes[Dy=0.1]{->}(0,0)(-4.5,0)(5.5,0.5) + \psset{linewidth=1pt,plotpoints=100} + \psGauss[mue=0,sigma=1]{-4.5}{4.5} + \psTDist[linecolor=blue]{-4}{4} + \psTDist[linecolor=red,nue=4]{-4}{4} +\end{pspicture} +\end{LTXexample} + +%The $t_\nu$-distribution has mode 0. + +\clearpage +\subsection{$F$-distribution} +A continuous statistical distribution which arises in the testing of +whether two observed samples have the same variance. + +The $F$-distribution with parameters $\mu$ and $\nu$ has the probability function +\[ +f_{\mu,\nu}(x)=\frac{\Gamma[(\mu+\nu)/2]}{\Gamma(\mu/2)\Gamma(\nu/2)}\cdot + \left(\mu/\nu\right)^{\mu/2}\frac{x^{(\mu/2)-1}}{[1+(\mu x/\nu)]^{(\mu+\nu)/2}}\quad +\text{ for $x>0$ and $\mu$, $\nu>0$}\] +% +and the syntax + +\begin{BDef} +\Lcs{psFDist}\OptArgs\Largb{x0}\Largb{x1} +\end{BDef} +% +The default settings are $\mu=1$ and $\nu=1$. + +\begin{LTXexample}[pos=t,preset=\centering] +\psset{xunit=2cm,yunit=10cm,plotpoints=100} +\begin{pspicture*}(-0.5,-0.07)(5.5,0.8) + \psline[linestyle=dashed](0.5,0)(0.5,0.75) + \psline[linestyle=dashed](! 2 7 div 0)(! 2 7 div 0.75) + \psset{linewidth=1pt} + \psFDist{0.1}{5} + \psFDist[linecolor=red,nue=3,mue=12]{0.01}{5} + \psFDist[linecolor=blue,nue=12,mue=3]{0.01}{5} + \psaxes[Dy=0.1]{->}(0,0)(5,0.75) +\end{pspicture*} +\end{LTXexample} + +\clearpage +\subsection{Beta distribution} + +A general type of statistical distribution which is related to the gamma distribution. +Beta distributions have two free parameters, which are labeled according to one of two +notational conventions. The usual definition calls these $\alpha$ and $\beta$, and the other +uses $\beta^\prime=\beta-1$ and $\alpha^\prime=\alpha-1$. The beta distribution is +used as a prior distribution for binomial proportions in \Index{Bayesian analysis}. +% +%The plots are for various values of ($\alpha,\beta$) with $\alpha=1$ and $\beta$ ranging from 0.25 to 3.00. +% +The domain is $[0,1]$, and the probability function $P(x)$ is given by +% +\[ +P(x) = \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}(1-x)^{\beta-1}x^{\alpha-1} +\quad\text{ $\alpha,\beta>0$} +\] +% +and has the syntax (with a default setting of $\alpha=1$ and $\beta=1$): + +\begin{BDef} +\Lcs{psBetaDist}\OptArgs\Largb{x0}\Largb{x1} +\end{BDef} +% +\begin{LTXexample}[pos=t,preset=\centering] +\psset{xunit=10cm,yunit=5cm} +\begin{pspicture*}(-0.1,-0.1)(1.1,2.05) + \psset{linewidth=1pt} + \multido{\rbeta=0.25+0.25,\ired=0+5,\rblue=50.0+-2.5}{20}{% + \psBetaDist[beta=\rbeta,linecolor=red!\ired!blue!\rblue]{0.01}{0.99}} + \psaxes[Dy=0.2,Dx=0.1]{->}(0,0)(1,2.01) +\end{pspicture*} +\end{LTXexample} + +\clearpage +\subsection{Cauchy distribution} +The \Index{Cauchy distribution}, also called the \Index{Lorentz distribution}, is a continuous distribution +describing resonance behavior. It also describes the distribution of horizontal distances at +which a line segment tilted at a random angle cuts the $x$-axis. + +The general Cauchy distribution and its cumulative distribution can be written as +\begin{align} +P(x) &= \frac{1}{\pi} \frac{b}{\left(x-m\right)^2+b^2}\\ +D(x) &= \frac12 +\frac{1}{\pi} \arctan\left(\frac{x-m}{b}\right) +\end{align} + +where \Lkeyword{b} is the half width at half maximum and \Lkeyword{m} is the statistical median. +The macro has the syntax (with a default setting of $m=0$ and $b=1$): + +\begin{BDef} +\Lcs{psCauchy}\OptArgs\Largb{x0}\Largb{x1}\\ +\Lcs{psCauchyI}\OptArgs\Largb{x0}\Largb{x1}\\ +\end{BDef} + +\Lcs{psCauchyI} is the integral or the cumulative distribution and often named as $D(x)$. + +\begin{LTXexample}[pos=t,preset=\centering] +\psset{xunit=2,yunit=3cm} +\begin{pspicture*}(-3,-0.3)(3.1,2.1) +\psset{linewidth=1pt} +\multido{\rb=0.1+0.2,\rm=0.0+0.2}{4}{% + \psCauchy[b=\rb,m=\rm,linecolor=red]{-2.5}{2.5} + \psCauchyI[b=\rb,m=\rm,linecolor=blue]{-2.5}{2.5}} +\psaxes[Dy=0.4,dy=0.4,Dx=0.5,dx=0.5]{->}(0,0)(-3,0)(3,2) +\end{pspicture*} +\end{LTXexample} +\iffalse + +\clearpage +\subsection{Bose-Einstein distribution} +A distribution which arises in the study of integer \Index{spin particles} in physics, +\[ +P(x)=\frac{x^s}{e^{x-\mu}-1}\qquad\text{with $s\in\mathbb{Z}$ and $\mu\in\mathbb{R}$} +\] +%$ +and has the syntax (with a default setting of $s=1$ and $\mu=1$): + +\begin{BDef} +\Lcs{psBoseEinsteinDist}\OptArgs\Largb{x0}\Largb{x1} +\end{BDef} +\fi + +\clearpage +\subsection{Weibull distribution} +In probability theory and statistics, the Weibull distribution is a continuous probability +distribution. The probability density function of a Weibull random variable $x$ is: + +\begin{align} +P(x) &= \alpha\beta^{-\alpha} x^{\alpha-1} e^{-\left(\frac{x}{\beta}\right)^\alpha}\\ +D(x) &= 1-e^{-\left(\frac{x}{\beta}\right)^\alpha} +\end{align} + +or slightly different as + +\begin{align} +P(x) &= \frac{\alpha}{\beta}\,x^{\alpha-1} e^{-\frac{x^\alpha}{\beta}}\\ +D(x) &= 1 - e^{-\frac{x^\alpha}{\beta}} +\end{align} + +always for $x\in[0;\infty)$, where $\alpha > 0$ is the shape parameter +and $\beta > 0$ is the scale parameter of the distribution. + +$D(x)$ is the cumulative distribution function of the Weibull distribution. The values for +$\alpha$ and $\beta$ are preset to 1, but can be changed in the usual way. + +The Weibull distribution is related to a number of other probability distributions; in +particular, it interpolates between the exponential distribution $(\alpha = 1)$ and +the Rayleigh distribution $(\alpha = 2)$. + +\begin{center} +\psset{unit=2} +\begin{pspicture*}(-0.5,-0.5)(2.6,2.6) +\psaxes{->}(0,0)(2.5,2.5)[$x$,-90][$y$,180] +\multido{\rAlpha=0.5+0.5}{5}{% + \psWeibull[alpha=\rAlpha]{0}{2.5} + \psWeibullI[alpha=\rAlpha,linestyle=dashed]{0}{2.4}} +\end{pspicture*} +% +\begin{pspicture*}(-0.5,-0.5)(2.6,2.6) +\psaxes{->}(0,0)(2.5,2.5)[$x$,-90][$y$,180] +\multido{\rAlpha=0.5+0.5,\rBeta=0.2+0.2}{5}{% + \psWeibull[alpha=\rAlpha,beta=\rBeta]{0}{2.5} + \psWeibullI[alpha=\rAlpha,beta=\rBeta,linestyle=dashed]{0}{2.4}} +\end{pspicture*} +\end{center} + +\begin{lstlisting} +\psset{unit=2} +\begin{pspicture*}(-0.5,-0.5)(2.6,2.6) +\psaxes{->}(0,0)(2.5,2.5)[$x$,-90][$y$,180] +\multido{\rAlpha=0.5+0.5}{5}{% + \psWeibull[alpha=\rAlpha]{0}{2.5} + \psWeibullI[alpha=\rAlpha,linestyle=dashed]{0}{2.4}} +\end{pspicture*} +% +\begin{pspicture*}(-0.5,-0.5)(2.6,2.6) +\psaxes{->}(0,0)(2.5,2.5)[$x$,-90][$y$,180] +\multido{\rAlpha=0.5+0.5,\rBeta=0.2+0.2}{5}{% + \psWeibull[alpha=\rAlpha,beta=\rBeta]{0}{2.5} + \psWeibullI[alpha=\rAlpha,beta=\rBeta,linestyle=dashed]{0}{2.4}} +\end{pspicture*} +\end{lstlisting} +\psset{unit=1cm} + +The starting value for $x$ should always be 0 or greater, if it is +less than 0 then the macro draws a line from (\#1,0) to (0,0) and +starts \Lcs{psWeinbull} with 0. + +\clearpage +\subsection{Vasicek distribution} + +For a homogenous portfolio of infinite granularity the portfolio loss +distribution is given by +\[ +\mathbb{P}(L(P)<x)=1-\mathcal{N} + \left(\frac{\mathcal{N}^{-1}(PD)-\sqrt{1-R2}\cdot\mathcal{N}^{-1}(x)}{R} + \right) +\] +$L(P)$ denotes the portfolio loss in percent, $pd$ is the uniform default +probability, and $R2$ is the uniform asset correlation. + +They are preset to $pd=0.22$ and $R2=0.11$ and can be overwritten in the +usual way. The macro uses the PostScript function norminv from the package +\LPack{pst-math} which is loaded by default and also shown in the following +example. + +\begin{LTXexample}[pos=t] +\psset{xunit=5} +\begin{pspicture}(-0.1,-3)(1.1,4) +\psaxes{->}(0,0)(0,-3)(1.1,4) +\psVasicek[plotpoints=200,linecolor=blue]{0}{0.9999} +\psVasicek[plotpoints=200,linecolor=red,pd=0.2,R2=0.3]{0}{0.9999} +\psplot[plotpoints=200,algebraic,linestyle=dashed]{0}{0.9999}{norminv(x)} +\end{pspicture} +\end{LTXexample} + +\clearpage +\section{The Lorenz curve} +The so-called \Index{Lorenz curve} is used in economics to describe inequality in +wealth or size. The Lorenz curve is a function of the cumulative proportion of +\textit{ordered individuals} mapped onto the corresponding cumulative proportion +of their size. Given a sample of $n^{\textup{th}}$ ordered individuals with +$x_i^{\prime}$ the size of individual $i$ and $x_1^{\prime}<x_2^{\prime}<\cdots<x_n^{\prime}$, +then the sample Lorenz curve is the \textit{polygon} joining the points $(h/n,L_h/L_n)$, +where $h=0, 1, 2,\ldots n, L_0=0$ and $L_h=\sum_{i=1}^h x_i^{\prime}$. + +\begin{BDef} +\LcsStar{psLorenz}\OptArgs\Largb{data file} +\end{BDef} + +\begin{LTXexample}[pos=t,preset=\centering] +\psset{lly=-6mm,llx=-5mm} +\psgraph[Dx=0.2,Dy=0.2,axesstyle=frame](0,0)(1,1){6cm}{6cm} +\psline[linestyle=dashed](1,1) +\psLorenz*[linecolor=blue!30,linewidth=1.5pt]{0.50 0.10 0.3 0.09 0.01 } +\psLorenz[linecolor=blue!30,plotstyle=bezier]{0.50 0.10 0.3 0.09 0.01 } +\psLorenz[linecolor=red,linewidth=1.5pt]{0.50 0.10 0.3 0.09 0.01 } +\endpsgraph +\end{LTXexample} + +There exists an optional argument \Lkeyword{Gini} for the output of the \Index{Gini coefficient}. +It is by default set to \false. With \true the value is caculated and printed below the +origin of the coordinate system. + +\begin{LTXexample}[pos=t,preset=\centering] +\psset{lly=-13mm,llx=-5mm} +\psgraph[Dx=0.2,Dy=0.2,axesstyle=frame](0,0)(1,1){6cm}{6cm} +\psline[linestyle=dashed](1,1) +\psLorenz[linewidth=1.5pt,Gini]{0.025 0.275 0.2 0.270 0.230} +\psLorenz[plotstyle=dots,dotstyle=square,dotscale=1.5]{0.025 0.275 0.2 0.270 0.230} +\endpsgraph +\end{LTXexample} + +\clearpage +\section{\nxLcs{psLame} -- Lam\'e Curve, a superellipse} +A superellipse is a curve with Cartesian equation +% +\begin{align} +\left|\frac{x}{a}\right|^r + \left|\frac{y}{b}\right|^r =1 +\end{align} +% +first discussed in 1818 by Gabriel Lam\'e (1795--1870)% +\footnote{Lam\'e worked on a wide variety of different topics. +His work on differential geometry and contributions to Fermat's Last Theorem +are important. He proved the theorem for $n = 7$ in 1839.}. +A superellipse may be described parametrically by +% +\begin{align} +x &= a\cdot\cos^{\frac{2}{r}} t\\ +y &= b\cdot\sin^{\frac{2}{r}} t +\end{align} +% +\Index{Superellipses} with $a=b$ are also known as \Index{Lam\'e} curves +or Lam\'e ovals and the restriction to $r>2$ is sometimes also made. +The following table summarizes a few special cases. +\Index{Piet Hein} used $\frac{5}{2}$ with a number of different +$\frac{a}{b}$ ratios for various of his projects. +For example, he used $\frac{a}{b}=\frac{6}{5}$ for Sergels Torg +(Sergel's Square) in Stockholm, and $\frac{a}{b}=\frac{3}{2}$ for his table. + +\begin{center} +\begin{tabular}{@{}llm{1.5cm}@{}} +r & curve type & example\\\hline +$\frac{2}{3}$ & (squashed) astroid + & \pspicture(-.5,-.5)(.5,.5)\psLame[radiusA=.5,radiusB=.5]{0.6667}\endpspicture\\ +1 & (squashed) diamond + & \pspicture(-.5,-.5)(.5,.5)\psLame[radiusA=.5,radiusB=.5]{1}\endpspicture\\ +2 & ellipse + & \pspicture(-.5,-.5)(.5,.5)\psLame[radiusA=.5,radiusB=.5]{2}\endpspicture\\ +$\frac{5}{2}$ & Piet Hein's ,,superellipse`` + & \pspicture(-.5,-.5)(.5,.5)\psLame[radiusA=.5,radiusB=.5]{2.5}\endpspicture +\end{tabular} +\end{center} + +If $r$ is a rational, then a \Index{superellipse} is algebraic. However, for irrational $r$, +it is transcendental. For even integers $r=n$, the curve becomes closer to a +rectangle as $n$ increases. The syntax of the \Lcs{psLame} macro is: + +\begin{BDef} +\Lcs{psLame}\OptArgs\Largb{r} +\end{BDef} + +It is internally plotted as a \Index{parametric plot} with $0\le\alpha\le360$. Available keywords +are \Lkeyword{radiusA} and \Lkeyword{radiusB}, both are preset to 1, but can have any valid value +and unit. + +\bgroup +\begin{LTXexample}[pos=t,preset=\centering] +\definecolorseries{col}{rgb}{last}{red}{blue} +\resetcolorseries[41]{col} +\psset{unit=.5} +\pspicture(-9,-9)(9,9) + \psaxes[Dx=2,Dy=2,tickstyle=bottom,ticksize=2pt]{->}(0,0)(-9,-9)(9,9) + \multido{\rA=0.2+0.1,\iA=0+1}{40}{% + \psLame[radiusA=8,radiusB=7,linecolor={col!![\iA]},linewidth=.5pt]{\rA}} +\endpspicture +\end{LTXexample} +\egroup + +\clearpage +\section{\nxLcs{psThomae} -- the popcorn function} +\Index{Thomae's function}, also known as the \Index{popcorn function}, +the \Index{raindrop function}, the \Index{ruler function} or the +\Index{Riemann function}, is a modification of the \Index{Dirichlet} function. +This real-valued function $f(x)$ is defined as follows: +% +\[ f(x)=\begin{cases} + \frac{1}{q}\mbox{ if }x=\frac{p}{q}\mbox{ is a rational number}\\ + 0\mbox{ if }x\mbox{ is irrational} + \end{cases} +\] +% +It is assumed here that $\mathop{gcd}(p,q) = 1$ and $q > 0$ so that the function is well-defined +and nonnegative. The syntax is: + +\begin{BDef} +\Lcs{psThomae}\OptArgs\Largr{x0,x1}\Largb{points} +\end{BDef} + +\verb+(x0,x1)+ is the plotted interval, both values must be grater zero and $x_1>x_0$. +The plotted number of points is the third parameter. + +\begin{LTXexample}[width=6cm,wide=false] +\psset{unit=4cm} +\begin{pspicture}(-0.1,-0.2)(2.5,1.15) + \psaxes{->}(0,0)(2.5,1.1) + \psThomae[dotsize=2.5pt,linecolor=red](0,2){300} +\end{pspicture} +\end{LTXexample} + + +\clearpage +\section{\nxLcs{psWeierstrass} -- a pathological function} + +The Weierstrass function is an example of a pathological real-valued function +on the real line. The function has the property that it is continuous +everywhere but differentiable nowhere. +% +\[ + f_a(x)=\sum\limits_{k=1}^\infty\frac{\sin(\pi k^ax)}{\pi k^a} +\] +%f(p/q)=pi/(4q^2)sum_(k=1)^(q-1)(sin((k^2ppi)/q))/(sin^2((kpi)/(2q))) +% + +\begin{BDef} +\Lcs{psWeierstrass}\OptArgs\Largr{$x_0,x_1$}\OptArg*{\Largs{a}}\Largb{a/b} +\end{BDef} + +Without the optional argument the mandatory one is $a$, otherwise it is $b$ and +the optional one $a$. Without setting the optional argument \Lkeyword{epsilon} the value +of 1.e-8 will be used. + + +\begin{LTXexample}[width=6.5cm,wide=false] +\psset{yunit=10,xunit=5} +\begin{pspicture}(-0.1,-0.5)(2.1,0.5) +\psaxes[Dx=0.2,Dy=0.1,ticksize=-2pt 0, + labelFontSize=\scriptstyle]{->}(0,0)(0,-0.5)(2.1,0.5) +\psWeierstrass[linecolor=red](0,2){2} +\psWeierstrass[linecolor=green,epsilon=1.e-15](0,2){3} +\psWeierstrass[linecolor=blue,epsilon=1.e-5](0,2){4} +\end{pspicture} +\end{LTXexample} + + +The original Weierstraß function can be used with the optional argument: +\[ f(x)= \sum_{n=0}^\infty a^n \cos(b^n \pi x) \] + + + +\begin{LTXexample}[width=6.5cm,wide=false] +\psset{unit=2cm,linewidth=0.5pt,plotpoints=5000} +\begin{pspicture}(-2.1,-2.1)(2.1,2.1) +\psaxes[Dx=0.5,Dy=0.5,ticksize=-2pt 0, + labelFontSize=\scriptstyle]{->}(0,0)(-2,-2)(2,2) +\psWeierstrass[linecolor=red](-2,2)[0.5]{3} +\psWeierstrass[linecolor=blue!70](-2,2)[0.5]{10} +\end{pspicture} +\end{LTXexample} + + + + +\clearpage +\section{\nxLcs{psplotImp} -- plotting implicit defined functions} +For a given area, the macro calculates in a first step row by row for every pixel (1pt) +the function $f(x,y)$ and checks for a changing of the value from $f(x,y)<0$ to $f(x,y)>0$ +or vice versa. If this happens, then the pixel must be part of the curve of +the function $f(x,y)=0$. In a second step the same is done column by column. +This may take some time because an area of $400\times 300$ pixel needs 120 thousand calculations +of the function value. The user still defines this area in his own coordinates, +the translation into pixel (pt) is done internally by the macro itself. +The only special keyword is \Lkeyword{stepFactor} which is preset to 0.67 and controls the horizontal +and vertical step width. + +\begin{BDef} +\Lcs{psplotImp}\OptArgs\Largr{xMin,yMin}\Largr{xMax,yMax}\OptArg{PS code}\Largb{function f(x,y)} +\end{BDef} + +The function must be of $f(x,y)=0$ and described in \PS code, or alternatively with +the option \Lkeyword{algebraic} (\LPack{pstricks-add}) in an algebraic form. +No other value names than $x$ and $y$ are possible. In general, a starred \Lenv{pspicture*} environment +maybe a good choice here. + +\medskip +\noindent +\begin{tabularx}{\linewidth}{!{\color{Orange!85!Red}\vrule width 5pt} X @{}} +The given area for \Lcs{psplotImp} should be \textbf{greater} than the given \Lenv{pspicture} area +(see examples). +\end{tabularx} + +\begin{LTXexample}[preset=\centering] +\begin{pspicture*}(-3,-3.2)(3.5,3.5) +\psaxes{->}(0,0)(-3,-3)(3.2,3)% +\psplotImp[linewidth=2pt,linecolor=red](-5,-2.1)(5,2.1){ x dup mul y dup mul add 4 sub } +\uput[45](0,2){$x^2+y^2-4=0$} +\psplotImp[linewidth=2pt,linecolor=blue,algebraic](-5,-3)(4,2.4){ (x+1)^2+y^2-4 } +\end{pspicture*} +\end{LTXexample} + +\begin{LTXexample}[preset=\centering] +\begin{pspicture*}(-3,-2.2)(3.5,2.5) +\psaxes{->}(0,0)(-3,-2)(3.2,2)% +\psplotImp[linewidth=2pt,linecolor=blue](-5,-2.2)(5,2.4){% + /xqu x dup mul def + /yqu y dup mul def + xqu yqu add dup mul 2 dup add 2 mul xqu yqu sub mul sub } +\uput*[0](-3,2){$\left(x^2+y^2\right)^2-8(x^2-y^2)=0$} +\psplotImp[linewidth=1pt,linecolor=red,algebraic](-5,-2.2)(5,2.4){% Lemniskate a =2 + (x^2+y^2)^2-4*(x^2-y^2) } +\end{pspicture*} +\end{LTXexample} + +\begin{LTXexample}[preset=\centering] +\begin{pspicture*}(-3,-3.2)(3.5,3.5) +\psaxes{->}(0,0)(-3,-3)(3.2,3)% +\psplotImp[linewidth=2pt,linecolor=green](-6,-6)(4,2.4){% + x 3 exp y 3 exp add 4 x y mul mul sub } +\uput*[45](-2.5,2){$\left(x^3+y^3\right)-4xy=0$} +\end{pspicture*} +\end{LTXexample} + +\begin{LTXexample}[preset=\centering] +\begin{pspicture*}(-5,-3.2)(5.5,4.5) +\psaxes{->}(0,0)(-5,-3)(5.2,4)% +\psplotImp[algebraic,linecolor=red](-6,-4)(5,4){ y*cos(x*y)-0.2 } +\psplotImp[algebraic,linecolor=blue](-6,-4)(5,4){ y*cos(x*y)-1.2 } +\end{pspicture*} +\end{LTXexample} + +Using the \Lkeyword{polarplot} option implies using the variables $r$ and $phi$ for describing +the function, $y$ and $x$ are not respected in this case. Using the \Lkeyword{algebraic} option +for polar plots are also possible (see next example). + +\begin{LTXexample}[preset=\centering] +\begin{pspicture*}(-3,-2.5)(3.75,2.75)\psaxes{->}(0,0)(-3,-2.5)(3.2,2.5)% +\psplotImp[linewidth=2pt,linecolor=cyan,polarplot](-6,-3)(4,2.4){ r 2 sub }% circle r=2 +\uput*[45](0.25,2){$f(r,\phi)=r-2=0$} +\psplotImp[polarplot,algebraic](-6,-3)(4,2.4){ r-1 }% circle r=1 +\end{pspicture*} +\end{LTXexample} + +\begin{LTXexample}[preset=\centering] +\begin{pspicture*}(-5,-2.2)(5.5,3.5) +\pscircle(0,0){1}% +\psaxes{->}(0,0)(-5,-2)(5.2,3)% +\multido{\rA=0.01+0.2}{5}{% +\psplotImp[linewidth=1pt,linecolor=blue,polarplot](-6,-6)(5,2.4){% + r dup mul 1.0 r div sub phi sin dup mul mul \rA\space sub }}% +\uput*[45](0,2){$f(r,\phi)=\left(r^2-\frac{1}{r}\right)\cdot\sin^2\phi=0$} +\end{pspicture*} +\end{LTXexample} + +\begin{LTXexample}[preset=\centering] +\begin{pspicture*}(-4,-3.2)(4.5,4.5) +\psaxes{->}(0,0)(-4,-3)(4.2,4)% +\psplotImp[algebraic,polarplot,linecolor=red](-5,-4)(5,4){ r+cos(phi/r)-2 } +\end{pspicture*} +\end{LTXexample} + +The data of an implicit plot can be written into an external file for further purposes. +Use the optional argument \Lkeyword[pstricks-add]{saveData} to write the $x|y$ values +into the file \nxLcs{jobname.data}. The file name can be changed with +the keyword \Lkeyword[pstricks-add]{filename}. When running a \TeX\ file from within a GUI +it may be possible that you get a writeaccess error from GhostScript, because it prevents writing +into a file when called from another program. In this case run GhostScript on the \PS-output from +the command line. + +\psset{mathLabel} +\begin{LTXexample}[preset=\centering] +\begin{pspicture*}(-3,-3)(3,3) + \psaxes[linewidth=0.25pt, + xlabelPos=top, + labelFontSize=\scriptscriptstyle, + labelsep=2pt, + ticksize=0.05]{<->}(0,0)(-2,-1.75)(2,2)[x,0][y,90] + \psplotImp[linecolor=red,linewidth=1pt,stepFactor=0.2,saveData, + algebraic](-2.5,-1.75)(2.5,2.5){x^2+(5*y/4-sqrt(abs(x)))^2-2.5} +\end{pspicture*} +\end{LTXexample} + +The values are saved pairwise in an array, e.\,g.: +\begin{verbatim} +... +[ +-1.53237 0.695058 +-1.53237 1.29957 +] +[ +-1.52534 0.666941 +-1.52534 1.32065 +] +... +\end{verbatim} + +In one array all $y$ values for the same $x$ value are stored. + +\iffalse +The data then can be read back to get a continous line of the plot. + +\begin{LTXexample}[preset=\centering] +\readdata[nStep=20]{\data}{\jobname.data} +\begin{pspicture*}(-3,-3)(3,3) + \psaxes[linewidth=0.25pt, + xlabelPos=top, + labelFontSize=\scriptscriptstyle, + labelsep=2pt, + ticksize=0.05]{<->}(0,0)(-2,-1.75)(2,2)[x,0][y,90] + \pslistplot[linecolor=red,linewidth=1pt,plotstyle=curve]{\data} +\end{pspicture*} +\end{LTXexample} +\fi + +\clearpage +\section{\nxLcs{psVolume} -- Rotating functions around the x-axis} +This macro shows the behaviour of a \Index{rotated function} around the $x$-axis. + +\begin{BDef} +\Lcs{psVolume}\OptArgs\Largr{xMin,xMax}\Largb{steps}\Largb{function $f(x)$} +\end{BDef} + +$f(x)$ has to be described as usual for the macro \Lcs{psplot}. + +\makebox[\linewidth]{% +\begin{pspicture}(-0.5,-2)(5,2.5) +\psaxes{->}(0,0)(0,-2)(3,2.5) +\psVolume[fillstyle=solid,fillcolor=magenta!30](0,4){1}{x sqrt} +\psline{->}(4,0)(5,0) +\end{pspicture} +% +\begin{pspicture}(-0.5,-2)(5,2.5) +\psaxes{->}(0,0)(0,-2)(3,2.5) +\psVolume[fillstyle=solid,fillcolor=red!40](0,4){2}{x sqrt} +\psline{->}(4,0)(5,0) +\end{pspicture} +% +\begin{pspicture}(-0.5,-2)(5,2.5) +\psaxes{->}(0,0)(0,-2)(3,2.5) +\psVolume[fillstyle=solid,fillcolor=blue!40](0,4){4}{x sqrt} +\psline{->}(4,0)(5,0) +\end{pspicture} +} + +\makebox[\linewidth]{% +\begin{pspicture}(-0.5,-2)(5,2.5) +\psaxes{->}(0,0)(0,-2)(3,2.5) +\psVolume[fillstyle=solid,fillcolor=green!40](0,4){8}{x sqrt} +\psline{->}(4,0)(5,0) +\end{pspicture} +% +\begin{pspicture}(-0.5,-2)(5,2.5) +\psaxes{->}(0,0)(0,-2)(3,2.5) +\psVolume[fillstyle=solid,fillcolor=yellow!40](0,4){16}{x sqrt} +\psline{->}(4,0)(5,0) +\end{pspicture} +% +\begin{pspicture}(-0.5,-2)(5,2.5) +\psaxes{->}(0,0)(0,-2)(3,2.5) +\psVolume[fillstyle=solid,fillcolor=cyan!40](0,4){32}{x sqrt} +\psline{->}(4,0)(5,0) +\end{pspicture} +} + +\begin{lstlisting} +\begin{pspicture}(-0.5,-2)(5,2.5) +\psaxes{->}(0,0)(0,-2)(3,2.5) +\psVolume[fillstyle=solid,fillcolor=magenta!30](0,4){1}{x sqrt} +\psline{->}(4,0)(5,0) +\end{pspicture} +% +\begin{pspicture}(-0.5,-2)(5,2.5) +\psaxes{->}(0,0)(0,-2)(3,2.5) +\psVolume[fillstyle=solid,fillcolor=red!40](0,4){2}{x sqrt} +\psline{->}(4,0)(5,0) +\end{pspicture} +% +\begin{pspicture}(-0.5,-2)(5,2.5) +\psaxes{->}(0,0)(0,-2)(3,2.5) +\psVolume[fillstyle=solid,fillcolor=blue!40](0,4){4}{x sqrt} +\psline{->}(4,0)(5,0) +\end{pspicture} + +\begin{pspicture}(-0.5,-2)(5,2.5) +\psaxes{->}(0,0)(0,-2)(3,2.5) +\psVolume[fillstyle=solid,fillcolor=green!40](0,4){8}{x sqrt} +\psline{->}(4,0)(5,0) +\end{pspicture} +% +\begin{pspicture}(-0.5,-2)(5,2.5) +\psaxes{->}(0,0)(0,-2)(3,2.5) +\psVolume[fillstyle=solid,fillcolor=yellow!40](0,4){16}{x sqrt} +\psline{->}(4,0)(5,0) +\end{pspicture} +% +\begin{pspicture}(-0.5,-2)(5,2.5) +\psaxes{->}(0,0)(0,-2)(3,2.5) +\psVolume[fillstyle=solid,fillcolor=cyan!40](0,4){32}{x sqrt} +\psline{->}(4,0)(5,0) +\end{pspicture} +\end{lstlisting} + +\psset{xunit=2} +\makebox[\linewidth]{% +\begin{pspicture}(-0.5,-4)(3,4) + \psaxes{->}(0,0)(0,-4)(3,4) + \psVolume[fillstyle=solid,fillcolor=cyan!40](0,1){4}{x} + \psVolume[fillstyle=solid,fillcolor=yellow!40](1,2){4}{x dup mul} + \psline(2,0)(3,0) +\end{pspicture} +% +\begin{pspicture}(-0.5,-4)(3,4) + \psaxes{->}(0,0)(0,-4)(3,4) + \psVolume[fillstyle=solid,fillcolor=cyan!40](0,1){20}{x} + \psVolume[fillstyle=solid,fillcolor=yellow!40](1,2){20}{x dup mul} + \psline(2,0)(3,0) +\end{pspicture} +} +\begin{lstlisting} +\psset{xunit=2} +\begin{pspicture}(-0.5,-4)(3,4) + \psaxes{->}(0,0)(0,-4)(3,4) + \psVolume[fillstyle=solid,fillcolor=cyan!40](0,1){4}{x} + \psVolume[fillstyle=solid,fillcolor=yellow!40](1,2){4}{x dup mul} + \psline(2,0)(3,0) +\end{pspicture} +% +\begin{pspicture}(-0.5,-4)(3,4) + \psaxes{->}(0,0)(0,-4)(3,4) + \psVolume[fillstyle=solid,fillcolor=cyan!40](0,1){20}{x} + \psVolume[fillstyle=solid,fillcolor=yellow!40](1,2){20}{x dup mul} + \psline(2,0)(3,0) +\end{pspicture} +\end{lstlisting} + + +\clearpage +\section{Examples} +\subsection{Filling an area under a distribution curve} +\begin{LTXexample}[preset=\centering] +\psset{xunit=0.5cm,yunit=20cm,arrowscale=1.5} +\begin{pspicture}(-1,-0.1)(21,0.2) +\psChiIIDist[linewidth=1pt,nue=5]{0.01}{19.5} +\psaxes[labels=none,ticks=none]{->}(20,0.2) +\pscustom[fillstyle=solid,fillcolor=red!30]{% + \psChiIIDist[linewidth=1pt,nue=5]{8}{19.5}% + \psline(20,0)(8,0)} +\end{pspicture} +\end{LTXexample} + + +\subsection{An animation of a distribution} + +\psset{xunit=0.9cm,yunit=9cm} +\newcommand*\studentT[1]{% + \begin{pspicture}(-6,-0.1)(6,0.5) + \psaxes[Dy=0.1]{->}(0,0)(-5,0)(5.5,0.45)[$x$,0][$y$,90] + \pscustom[fillstyle=solid,fillcolor=blue!40,opacity=0.4,linecolor=red,linestyle=none]{% + \psline(0,0)(-5,0) + \psTDist[nue=#1]{-5}{5} + \psline(5,0)(0,0) + } + \psTDist[nue=#1,linecolor=red,linewidth=1pt]{-5}{5} + \rput(3,0.3){$\nu = #1$} + \end{pspicture}} + +\begin{center} + \begin{animateinline}[poster=first,controls,palindrome]{10} + \multiframe{50}{rA=0.02+0.02}{\studentT{\rA}} + \end{animateinline} + \captionof{figure}{Student's $t$-distribution.} +\end{center} + + +\begin{lstlisting} +\psset{xunit=0.9cm,yunit=9cm} +\newcommand*\studentT[1]{% + \begin{pspicture}(-6,-0.1)(6,0.5) + \psaxes[Dy=0.1]{->}(0,0)(-5,0)(5.5,0.45)[$x$,0][$y$,90] + \pscustom[fillstyle=solid,fillcolor=blue!40,opacity=0.4,linecolor=red,linestyle=none]{% + \psline(0,0)(-5,0) + \psTDist[nue=#1]{-5}{5} + \psline(5,0)(0,0) + } + \psTDist[nue=#1,linecolor=red,linewidth=1pt]{-5}{5} + \rput(3,0.3){$\nu = #1$} + \end{pspicture}} + +\begin{center} + \begin{animateinline}[poster=first,controls,palindrome]{10} + \multiframe{50}{rA=0.02+0.02}{\studentT{\rA}} + \end{animateinline} + \captionof{figure}{Student's $t$-distribution.} +\end{center} +\end{lstlisting} + + +\clearpage +\section{List of all optional arguments for \texttt{pst-func}} +\xkvview{family=pst-func,columns={key,type,default}} + +\bgroup +\RaggedRight +\nocite{*} +%\bibliographystyle{plain} +\printbibliography{pst-func-doc} +\egroup + +\printindex + +\end{document} |